PL9fwy3NUQKwaIc7KWc8buW344941GStQL/-VA6U2qwaG0_raw.srt

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Last time we have discussed hypothesis test for

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two population proportions. And we mentioned that

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the assumptions are for the first sample. n times

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pi should be at least 5, and also n times 1 minus

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pi is also at least 5. The same for the second

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sample, n 2 times pi 2 is at least 5, as well as n

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times 1 minus pi 2 is also at least 5. Also, we

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discussed that the point estimate for the

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difference of Pi 1 minus Pi 2 is given by P1 minus

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P2. That means this difference is unbiased point

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estimate of Pi 1 minus Pi 2. Similarly, P2 minus

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P1 is the point estimate of the difference Pi 2

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minus Pi 1.

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We also discussed that the bold estimate for the

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overall proportion is given by this equation. So B

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dash is called the bold estimate for the overall

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proportion. X1 and X2 are the number of items of

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interest. And the two samples that you have in one

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and two, where in one and two are the sample sizes

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for the first and the second sample respectively.

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The appropriate statistic in this course is given

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by this equation. Z-score or Z-statistic is the

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point estimate of the difference pi 1 minus pi 2

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minus the hypothesized value under if 0, I mean if

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0 is true, most of the time this term equals 0,

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divided by this quantity is called the standard

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error of the estimate, which is square root of B

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dash 1 minus B dash times 1 over N1 plus 1 over

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N2. So this is your Z statistic. The critical

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regions. I'm sorry, first, the appropriate null

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and alternative hypothesis are given by three

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cases we have. Either two-tailed test or one

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-tailed and it has either upper or lower tail. So

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for example, for lower-tailed test, We are going

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to test to see if a proportion 1 is smaller than a

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proportion 2. This one can be written as pi 1

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smaller than pi 2 under H1, or the difference

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between these two population proportions is

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negative, is smaller than 0. So either you may

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write the alternative as pi 1 smaller than pi 2,

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or the difference, which is pi 1 minus pi 2

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smaller than 0. For sure, the null hypothesis is

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the opposite of the alternative hypothesis. So if

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this is one by one smaller than by two, so the

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opposite by one is greater than or equal to two.

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Similarly, but the opposite side here, we are

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talking about the upper tail of probability. So

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under the alternative hypothesis, by one is

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greater than by two. Or it could be written as by

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one minus by two is positive, that means greater

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than zero. While for the two-tailed test, for the

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alternative hypothesis, we have Y1 does not equal

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Y2. In this case, we are saying there is no

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difference under H0, and there is a difference.

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should be under each one. Difference means either

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greater than or smaller than. So we have this not

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equal sign. So by one does not equal by two. Or it

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could be written as by one minus by two is not

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equal to zero. It's the same as the one we have

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discussed when we are talking about comparison of

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two population means. We just replaced these by's

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by mus. Finally, the rejection regions are given

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by three different charts here for the lower tail

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test. We reject the null hypothesis if the value

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of the test statistic fall in the rejection

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region, which is in the left side. So that means

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we reject zero if this statistic is smaller than

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negative zero. That's for lower tail test. On the

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other hand, for other tailed tests, your rejection

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region is the right side, so you reject the null

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hypothesis if this statistic is greater than Z

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alpha. In addition, for two-tailed tests, there

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are two rejection regions. One is on the right

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side, the other on the left side. Here, alpha is

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split into two halves, alpha over two to the

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right, similarly alpha over two to the left side.

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Here, we reject the null hypothesis if your Z

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statistic falls in the rejection region here, that

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means z is smaller than negative z alpha over 2 or

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z is greater than z alpha over 2. Now this one, I

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mean the rejection regions are the same for either

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one sample t-test or two sample t-test, either for

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the population proportion or the population mean.

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We have the same rejection regions. Sometimes we

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replace z by t. It depends if we are talking about

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small samples and sigmas unknown. So that's the

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basic concepts about testing or hypothesis testing

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for the comparison between two population

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proportions. And we stopped at this point. I will

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give three examples, three examples for testing

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about two population proportions. The first one is

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given here. It says that, is there a significant

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difference between the proportion of men and the

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proportion of women who will vote yes on a

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proposition?

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In this case, we are talking about a proportion.

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So this problem tests for a proportion. We have

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two proportions here because we have two samples

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for two population spheres, men and women. So

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there are two populations. So we are talking about

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two population proportions. Now, we have to state

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carefully now an alternative hypothesis. So for

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example, let's say that phi 1 is the population

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proportion, proportion of men who will vote for a

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proposition A for example, for vote yes, for vote

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yes for proposition A.

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is the same but of men, of women, I'm sorry. So

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the first one for men and the other of

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women. Now, in a random, so in this case, we are

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talking about difference between two population

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proportions, so by one equals by two.

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Your alternate hypothesis should be, since the

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problem talks about, is there a significant

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difference? Difference means two tails. So it

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should be pi 1 does not equal pi 2. Pi 1 does not

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equal pi 2. So there's still one state null and

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alternate hypothesis. Now, in a random sample of

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36 out of 72 men, And 31 of 50 women indicated

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they would vote yes. So for example, if X1

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represents number of men who would vote yes, that

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means X1 equals 36 in

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172. So that's for men. Now for women. 31 out of

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50. So 50 is the sample size for the second

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sample. Now it's ask about this test about the

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difference between the two population proportion

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at 5% level of significance. So alpha is given to

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be 5%. So that's all the information you have in

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order to answer this question. So based on this

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statement, we state null and alternative

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hypothesis. Now based on this information, we can

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solve the problem by using three different

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approaches. Critical value approach, B value, and

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confidence interval approach. Because we can use

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confidence interval approach because we are

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talking about two-tailed test. So let's start with

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the basic one, critical value approach. So

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approach A.

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Now since we are talking about two-tailed test,

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your critical value should be plus or minus z

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alpha over 2. And since alpha is 5% so the

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critical values are z

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plus or minus 0.25 which is 196. Or you may use

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the standard normal table in order to find the

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critical values. Or just if you remember that

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values from previous time. So the critical regions

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are above 196 or smaller than negative 196. I have

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to compute the Z statistic. Now Z statistic is

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given by this equation. Z stat equals B1 minus B2.

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minus Pi 1 minus Pi 2. This quantity divided by P

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dash 1 minus P dash multiplied by 1 over N1 plus 1

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over N1. Here we have to find B1, B2. So B1 equals

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X1 over N1. X1 is given.

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to that means 50%. Similarly,

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B2 is A equals X2 over into X to the third power

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over 50, so that's 60%. Also, we have to compute

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the bold estimate of the overall proportion of B

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dash

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What are the sample sizes we have? X1 and X2. 36

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plus 31. Over 72 plus 7. 72 plus 7. So that means

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67 over 152.549.

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120.

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So simple calculations give B1 and B2, as well as

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B dash. Now, plug these values on the Z-state

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formula, we get the value that is this. So first,

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state null and alternative hypothesis, pi 1 minus

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pi 2 equals 0. That means the two populations are

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equal. We are going to test this one against Pi 1

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minus Pi 2 is not zero. That means there is a

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significant difference between proportions. Now

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for men, we got proportion of 50%. That's for the

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similar proportion. And similar proportion for

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women who will vote yes for position A is 62%. The

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pooled estimate for the overall proportion equals

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0.549. Now, based on this information, we can

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calculate the Z statistic. Straightforward

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calculation, you will end with this result. So, Z

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start negative 1.31.

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So, we have to compute this one before either

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before using any of the approaches we have.

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If we are going to use their critical value

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approach, we have to find Z alpha over 2 which is

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1 more than 6. Now the question is, is this value

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falling the rejection regions right or left? it's

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clear that this value, negative 1.31, lies in the

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non-rejection region, so we don't reject a null

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hypothesis. So my decision is don't reject H0. My

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conclusion is there is not significant evidence of

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a difference in proportions who will vote yes

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between men and women. Even it seems to me that

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there is a difference between Similar proportions,

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50% and 62%. Still, this difference is not

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significant in order to say that there is

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significant difference between the proportions of

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men and women. So based on the critical value

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approach. We end with this result, which is we

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don't reject null hypotheses. That means the

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information you have is not sufficient in order to

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support alternative hypotheses. So your managerial

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conclusion should be there is not significant

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difference in proportion and proportions who will

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vote yes between men and women. That's for using

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critical value approach. Before continue, we have

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to discuss the confidence interval for the

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difference pi 1 minus pi 2. The confidence

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interval, as we mentioned before, can be

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constructed by point estimate, plus or minus

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critical value times the standard error of the

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point estimate. In this case, the point estimate

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for pi 1 minus pi 2 is b1 minus b2. So that's your

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point estimate, plus or minus z alpha over 2. Now

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from the information from chapter 8, the standard

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error of the difference, b1 minus pi 2, is given

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by this equation. B1 times 1 minus B1, so B1 and

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its complement, divided by the first sample size,

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plus the second sample proportion times its

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complement divided by the sample size of the

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second sample. So that's your confidence interval.

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So it looks similar to the one we have discussed

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for the mu 1 minus mu 2. And that one we had x1

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bar minus x2 bar plus or minus z or t, it depends

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on the sample sizes, times s square b times 1 over

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n1 plus 1 over n2. Anyway, the confidence interval

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for pi 1 minus pi 2 is given by this equation. Now

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let's see how can we use the other two approaches

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in order to test if there is significant

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difference between the proportions of men and

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women. I'm sure you don't have this slide for

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computing B value and confidence interval.

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Now since we are talking about two-thirds, your B

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value should be the probability of Z greater than

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1.31 and smaller than negative 1.31. So my B value

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in this case equals Z greater than 1.31 plus Z

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smaller than negative 1.31. Since we are talking

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about two tail tests, so there are two rejection

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regions. My Z statistic is 1.31, so it should be

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here 1.31 to the right, and negative. Now, what's

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the probability that the Z statistic will fall in

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the rejection regions, right or left? So we have

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to add. B of Z greater than 1.31 and B of Z

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smaller than negative. Now the two areas to the

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right of 1.31 and to the left of negative 1.31 are

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equal because of symmetry. So just compute one and

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multiply that by two, you will get the B value. So

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two times. Now by using the concept in chapter

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six, easily you can compute either this one or the

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other one. The other one directly from the

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negative z-score table. The other one you should

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have the complement 1 minus, because it's smaller

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than 1.1. And either way you will get this result.

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Now my p-value is around 19%. Always we reject the

251
00:19:11,750 --> 00:19:14,930
null hypothesis. if your B value is smaller than

252
00:19:14,930 --> 00:19:20,410
alpha that always we reject null hypothesis if my

253
00:19:20,410 --> 00:19:25,950
B value is smaller than alpha alpha is given 5%

254
00:19:25,950 --> 00:19:31,830
since B value equals

255
00:19:31,830 --> 00:19:36,910
19% which is much bigger than Much greater than

256
00:19:36,910 --> 00:19:41,170
5%, so we don't reject our analysis. So my

257
00:19:41,170 --> 00:19:48,390
decision is we don't reject at zero. So the same

258
00:19:48,390 --> 00:19:52,690
conclusion as we reached by using critical

259
00:19:52,690 --> 00:19:57,850
penalty. So again, by using B value, we have to

260
00:19:57,850 --> 00:20:00,770
compute the probability that your Z statistic

261
00:20:00,770 --> 00:20:05,320
falls in the rejection regions. I end with this

262
00:20:05,320 --> 00:20:10,600
result, my B value is around 19%. As we mentioned

263
00:20:10,600 --> 00:20:14,180
before, we reject null hypothesis if my B value is

264
00:20:14,180 --> 00:20:17,920
smaller than alpha. Now, my B value in this case

265
00:20:17,920 --> 00:20:22,740
is much, much bigger than 5%, so my decision is

266
00:20:22,740 --> 00:20:26,860
don't reject null hypothesis. Any questions?

267
00:20:36,140 --> 00:20:41,160
The other approach, the third one, confidence

268
00:20:41,160 --> 00:20:42,520
interval approach.

269
00:20:46,260 --> 00:20:48,980
Now, for the confidence interval approach, we have

270
00:20:48,980 --> 00:20:53,960
this equation, b1 minus b2. Again, the point

271
00:20:53,960 --> 00:21:03,760
estimate, plus or minus z square root b1 times 1

272
00:21:03,760 --> 00:21:09,810
minus b1 divided by a1. B2 times 1 minus B2

273
00:21:09,810 --> 00:21:11,650
divided by N2.

274
00:21:13,850 --> 00:21:20,730
Now we have B1 and B2, so 0.5 minus 0.62. That's

275
00:21:20,730 --> 00:21:25,170
your calculations from previous information we

276
00:21:25,170 --> 00:21:28,470
have. Plus or minus Z alpha over 2, the critical

277
00:21:28,470 --> 00:21:35,030
value again is 1.96 times Square root of P1.5

278
00:21:35,030 --> 00:21:41,090
times 1 minus 0.5 divided by N1 plus P2 62 percent

279
00:21:41,090 --> 00:21:46,550
times 1 minus P2 divided by N2. 0.5 minus 62

280
00:21:46,550 --> 00:21:50,650
percent is negative 12 percent plus or minus the

281
00:21:50,650 --> 00:21:53,090
margin of error. This amount is again as we

282
00:21:53,090 --> 00:21:56,730
mentioned before is the margin of error is 0.177.

283
00:21:57,530 --> 00:21:59,830
Now simple calculation will end with this result

284
00:21:59,830 --> 00:22:03,300
that is The difference between the two proportions

285
00:22:03,300 --> 00:22:09,820
lie between negative 0.296 and 0.057. That means

286
00:22:09,820 --> 00:22:14,580
we are 95% confident that the difference between

287
00:22:14,580 --> 00:22:19,100
the proportions of men who will vote yes for a

288
00:22:19,100 --> 00:22:27,640
position A and men equals negative 0.297 up to 0

289
00:22:27,640 --> 00:22:31,680
.057. Now the question is since we are testing

290
00:22:31,680 --> 00:22:37,380
it's zero by one minus by two equals zero the

291
00:22:37,380 --> 00:22:41,700
question is does this interval contain zero or

292
00:22:41,700 --> 00:22:47,680
capture zero? Now since we start here from

293
00:22:47,680 --> 00:22:51,230
negative and end with positive, I mean the lower

294
00:22:51,230 --> 00:22:55,330
bound is negative 0.297 and the upper bound is 0

295
00:22:55,330 --> 00:23:00,610
.057. So zero inside the interval, I mean the

296
00:23:00,610 --> 00:23:03,870
confidence interval contains zero in this case, so

297
00:23:03,870 --> 00:23:06,650
we don't reject the null hypothesis because maybe

298
00:23:06,650 --> 00:23:11,780
the difference equals zero. So since this interval

299
00:23:11,780 --> 00:23:16,300
does contain the hypothesis difference zero, so we

300
00:23:16,300 --> 00:23:21,100
don't reject null hypothesis at 5% level. So the

301
00:23:21,100 --> 00:23:24,880
same conclusion as we got before by using critical

302
00:23:24,880 --> 00:23:27,460
value approach and de-value approach. So either

303
00:23:27,460 --> 00:23:32,100
one will end with the same decision. Either reject

304
00:23:32,100 --> 00:23:37,020
or fail to reject, it depends on the test itself.

305
00:23:38,760 --> 00:23:43,820
That's all. Do you have any question? Any

306
00:23:43,820 --> 00:23:47,540
question? So again, there are three different

307
00:23:47,540 --> 00:23:51,600
approaches in order to solve this problem. One is

308
00:23:51,600 --> 00:23:55,680
critical value approach, the standard one. The

309
00:23:55,680 --> 00:23:58,900
other two are the value approach and confidence

310
00:23:58,900 --> 00:24:02,140
interval. One more time, confidence interval is

311
00:24:02,140 --> 00:24:07,080
only valid for

312
00:24:08,770 --> 00:24:13,110
two-tailed test. Because the confidence interval

313
00:24:13,110 --> 00:24:16,430
we have is just for two-tailed test, so it could

314
00:24:16,430 --> 00:24:20,210
be used only for testing about two-tailed test.

315
00:24:23,350 --> 00:24:25,990
As we mentioned before, I'm going to skip

316
00:24:25,990 --> 00:24:32,390
hypothesis for variances as well as ANOVA test. So

317
00:24:32,390 --> 00:24:36,410
that's all for chapter ten.

318
00:24:37,670 --> 00:24:42,390
But now I'm going to do some of the practice

319
00:24:42,390 --> 00:24:43,730
problems.

320
00:24:46,750 --> 00:24:52,630
Chapter 10. To practice, let's start with some

321
00:24:52,630 --> 00:24:55,270
practice problems for Chapter 10.

322
00:24:59,270 --> 00:25:03,770
A few years ago, Pepsi invited consumers to take

323
00:25:03,770 --> 00:25:08,870
the Pepsi challenge. Consumers were asked to

324
00:25:08,870 --> 00:25:13,790
decide which of two sodas, Coke or Pepsi. They

325
00:25:13,790 --> 00:25:17,930
preferred an applied taste test. Pepsi was

326
00:25:17,930 --> 00:25:21,930
interested in determining what factors played a

327
00:25:21,930 --> 00:25:25,930
role in people's taste preferences. One of the

328
00:25:25,930 --> 00:25:28,630
factors studied was the gender of the consumer.

329
00:25:29,650 --> 00:25:32,350
Below are the results of the analysis comparing

330
00:25:32,350 --> 00:25:36,870
the taste preferences of men and women with the

331
00:25:36,870 --> 00:25:41,630
proportions depicting preference in or for Pepsi.

332
00:25:42,810 --> 00:25:49,190
For meals, size

333
00:25:49,190 --> 00:25:57,990
of 109. So that's your N1. And proportion.

334
00:26:00,480 --> 00:26:09,100
for males is around 4.2. For females,

335
00:26:11,640 --> 00:26:25,720
N2 equals 52, and proportion of females, 25%. The

336
00:26:25,720 --> 00:26:29,870
difference between proportions of men and women or

337
00:26:29,870 --> 00:26:35,590
males and females is 0.172, around 0.172. And this

338
00:26:35,590 --> 00:26:41,530
statistic is given by 2.118, so approximately 2

339
00:26:41,530 --> 00:26:47,170
.12. Now, based on this result, based on this

340
00:26:47,170 --> 00:26:49,090
information, question number one,

341
00:26:53,910 --> 00:26:58,690
To determine if a difference exists in the test

342
00:26:58,690 --> 00:27:04,490
preferences of men and women, give the correct

343
00:27:04,490 --> 00:27:06,970
alternative hypothesis that lives through a test.

344
00:27:08,830 --> 00:27:15,830
A. B. Why B? Because the test defines between the

345
00:27:15,830 --> 00:27:18,650
new form A and the new form B. Because if we say

346
00:27:18,650 --> 00:27:21,910
that H1 is equal to U1 minus M equals F,

347
00:27:28,970 --> 00:27:34,190
So the correct answer is B? B. So that's

348
00:27:34,190 --> 00:27:40,830
incorrect. C. Why? Why C is the correct answer?

349
00:27:45,470 --> 00:27:46,070
Because

350
00:27:52,720 --> 00:27:56,500
Y is not equal because we have difference. So

351
00:27:56,500 --> 00:27:59,380
since we have difference here, it should be not

352
00:27:59,380 --> 00:28:02,240
equal to. And since we are talking about

353
00:28:02,240 --> 00:28:06,120
proportions, so you have to ignore A and B. So A

354
00:28:06,120 --> 00:28:10,020
and B should be ignored first. Then you either

355
00:28:10,020 --> 00:28:15,220
choose C or D. C is the correct answer. So C is

356
00:28:15,220 --> 00:28:20,440
the correct answer. That's for number one. Part

357
00:28:20,440 --> 00:28:27,100
two. Now suppose Pepsi wanted to test to determine

358
00:28:27,100 --> 00:28:35,680
if males preferred Pepsi more than females. Using

359
00:28:35,680 --> 00:28:38,400
the test statistic given, compute the appropriate

360
00:28:38,400 --> 00:28:43,940
p-value for the test. Let's assume that pi 1 is

361
00:28:43,940 --> 00:28:48,640
the population proportion for males who preferred

362
00:28:48,640 --> 00:28:56,440
Pepsi, and pi 2 for females who prefer Pepsi. Now

363
00:28:56,440 --> 00:29:00,140
he asks about suppose the company wanted to test

364
00:29:00,140 --> 00:29:02,760
to determine if males prefer Pepsi more than

365
00:29:02,760 --> 00:29:08,080
females. Using again the statistic given which is

366
00:29:08,080 --> 00:29:13,400
2.12 for example, compute appropriately value. Now

367
00:29:13,400 --> 00:29:18,160
let's state first H0 and H8.

368
00:29:27,450 --> 00:29:31,970
H1 pi 1

369
00:29:31,970 --> 00:29:34,410
minus pi 2 is greater than zero.

370
00:29:37,980 --> 00:29:42,740
Because it says that males prefer Pepsi more than

371
00:29:42,740 --> 00:29:46,940
females. Bi-1 for males, Bi-2 for females. So I

372
00:29:46,940 --> 00:29:50,800
should have Bi-1 greater than Bi-2 or Bi-1 minus

373
00:29:50,800 --> 00:29:54,940
Bi-2 is positive. So it's upper case. Now, in this

374
00:29:54,940 --> 00:30:01,940
case, my B value, its probability is B.

375
00:30:05,680 --> 00:30:07,320
It's around this value.

376
00:30:12,410 --> 00:30:18,230
1 minus b of z is smaller than 2.12. So 1 minus,

377
00:30:18,350 --> 00:30:21,530
now by using the table or the z table we have.

378
00:30:25,510 --> 00:30:29,370
Since we are talking about 2, 1, 12, 2, 1, 2, I'm

379
00:30:29,370 --> 00:30:34,670
sorry, 2, 1, 2, 2, 1, 2, so the answer is 983. So

380
00:30:34,670 --> 00:30:40,590
1 minus 893, so the answer is 017. So my b value.

381
00:30:43,430 --> 00:30:49,890
equals 0 and 7. So A is the correct answer. Now if

382
00:30:49,890 --> 00:30:53,970
the problem is two-tailed test, it should be

383
00:30:53,970 --> 00:30:57,450
multiplied by 2. So the answer, the correct should

384
00:30:57,450 --> 00:31:02,230
be B. So you have A and B. If it is one-third,

385
00:31:02,390 --> 00:31:06,310
your correct answer is A. If it is two-thirds, I

386
00:31:06,310 --> 00:31:10,550
mean if we are testing to determine if a

387
00:31:10,550 --> 00:31:13,890
difference exists, then you have to multiply this

388
00:31:13,890 --> 00:31:19,030
one by two. So that's your B value. Any question?

389
00:31:23,010 --> 00:31:27,550
Number three. Suppose Babs wanted to test to

390
00:31:27,550 --> 00:31:33,230
determine if meals If males prefer Pepsi less than

391
00:31:33,230 --> 00:31:36,810
females, using the statistic given, compute the

392
00:31:36,810 --> 00:31:42,990
product B value. Now, H1 in this case, B1 is

393
00:31:42,990 --> 00:31:48,490
smaller than Z by 2, by 1 smaller than 1. Now your

394
00:31:48,490 --> 00:31:54,490
B value, Z is smaller than, because here it is

395
00:31:54,490 --> 00:31:58,050
smaller than my statistic 2.12.

396
00:32:01,570 --> 00:32:04,790
We don't write negative sign. Because the value of

397
00:32:04,790 --> 00:32:08,150
the statistic is 2.12. But here we are going to

398
00:32:08,150 --> 00:32:11,790
test lower tail test. So my B value is B of Z

399
00:32:11,790 --> 00:32:15,250
smaller than. So smaller comes from the

400
00:32:15,250 --> 00:32:17,730
alternator. This is the sign under the alternator.

401
00:32:18,910 --> 00:32:21,810
And you have to take the value of the Z statistic

402
00:32:21,810 --> 00:32:22,510
as it is.

403
00:32:25,610 --> 00:32:34,100
So B of Z is smaller than minus 3. so they need if

404
00:32:34,100 --> 00:32:38,060
you got a correct answer D is the correct if B is

405
00:32:38,060 --> 00:32:40,420
the correct answer you will get nine nine nine six

406
00:32:40,420 --> 00:32:47,620
six that's incorrect answer any question the

407
00:32:47,620 --> 00:32:53,920
correct is D number

408
00:32:53,920 --> 00:32:57,620
four suppose

409
00:32:57,620 --> 00:33:03,650
that Now for example, forget the information we

410
00:33:03,650 --> 00:33:07,390
have so far for B value. Suppose that the two

411
00:33:07,390 --> 00:33:11,910
-tailed B value was really 0734. Now suppose my B

412
00:33:11,910 --> 00:33:19,010
value for two-tailed is 0734. That's for two

413
00:33:19,010 --> 00:33:20,210
-tailed. This is my B value.

414
00:33:23,070 --> 00:33:28,490
This is my B value. It's 0, 7, 3, 4. Now we have

415
00:33:28,490 --> 00:33:33,650
four answers. Part A, B, C, and D. Which one is

416
00:33:33,650 --> 00:33:34,050
the correct?

417
00:33:42,030 --> 00:33:46,610
A says at 5% level, there is sufficient evidence

418
00:33:46,610 --> 00:33:51,510
to conclude the proportion of males Preferring

419
00:33:51,510 --> 00:33:53,930
Pepsi differs from the proportion of females

420
00:33:53,930 --> 00:33:58,970
preferring Pepsi. Which one is the correct answer?

421
00:34:02,290 --> 00:34:04,550
B value is 0.734.

422
00:34:10,370 --> 00:34:16,650
B it says at alpha equals 10 percent. There is

423
00:34:16,650 --> 00:34:20,320
sufficient evidence. to indicate the proportion of

424
00:34:20,320 --> 00:34:22,900
males preferring Pepsi differs from the proportion

425
00:34:22,900 --> 00:34:24,160
of females preferring Pepsi.

426
00:34:27,240 --> 00:34:30,720
C. At 5%, there is sufficient evidence to indicate

427
00:34:30,720 --> 00:34:33,260
the proportion of males preferring Pepsi equals

428
00:34:33,260 --> 00:34:38,860
the proportion of females preferring Pepsi. D. At

429
00:34:38,860 --> 00:34:42,480
8% level, there is insufficient evidence to

430
00:34:42,480 --> 00:34:45,860
include to indicate the proportion of males

431
00:34:45,860 --> 00:34:48,580
preferring babies differs from the proportion of

432
00:34:48,580 --> 00:34:49,720
females preferring babies.

433
00:34:54,300 --> 00:34:59,360
Again, suppose that here it's two-tailed test. It

434
00:34:59,360 --> 00:35:03,420
says two-tailed test. Two-tailed means Y1 does not

435
00:35:03,420 --> 00:35:09,540
equal Y2. So in this case, we are testing Y1

436
00:35:09,540 --> 00:35:16,190
equals Y2. against by one is not by two and your B

437
00:35:16,190 --> 00:35:19,090
value is zero seven three four. So which one is

438
00:35:19,090 --> 00:35:27,950
the correct answer? B? D. Let's look at D. Let's

439
00:35:27,950 --> 00:35:29,270
look at D.

440
00:35:34,610 --> 00:35:39,900
Since B value is smaller than alpha, Since it

441
00:35:39,900 --> 00:35:44,840
means we reject Insufficient means we don't reject

442
00:35:44,840 --> 00:35:50,380
So D is incorrect D

443
00:35:50,380 --> 00:35:52,440
is incorrect because here there is insufficient

444
00:35:52,440 --> 00:35:55,720
Since

445
00:35:55,720 --> 00:35:58,800
we if we reject it means that we have sufficient

446
00:35:58,800 --> 00:36:02,700
evidence so support The alternative so D is

447
00:36:02,700 --> 00:36:07,470
incorrect Now what's about C at five percent Five,

448
00:36:07,830 --> 00:36:10,570
so this value is greater than five, so we don't

449
00:36:10,570 --> 00:36:13,270
reject. So that's incorrect.

450
00:36:21,370 --> 00:36:28,030
B. At five, at 10% now, there is sufficient

451
00:36:28,030 --> 00:36:34,550
evidence. Sufficient means we reject. We reject.

452
00:36:35,220 --> 00:36:40,440
Since this B value, 0.7, is smaller than alpha. 7%

453
00:36:40,440 --> 00:36:44,240
is smaller than 10%. So we reject. That means you

454
00:36:44,240 --> 00:36:46,960
have to read carefully. There is sufficient

455
00:36:46,960 --> 00:36:50,280
evidence to include, to indicate the proportion of

456
00:36:50,280 --> 00:36:54,820
males preferring Pepsi differs from the proportion

457
00:36:54,820 --> 00:36:58,660
of females. That's correct. So B is the correct

458
00:36:58,660 --> 00:37:05,570
state. Now look at A. A, at 5% there is sufficient

459
00:37:05,570 --> 00:37:09,710
evidence? No, because this value is greater than

460
00:37:09,710 --> 00:37:16,970
alpha, so we don't reject. For this one. Here we

461
00:37:16,970 --> 00:37:21,050
reject because at 10% we reject. So B is the

462
00:37:21,050 --> 00:37:27,670
correct answer. Make sense? Yeah, exactly, for

463
00:37:27,670 --> 00:37:31,850
10%. If this value is 5%, then B is incorrect.

464
00:37:34,190 --> 00:37:38,690
Again, if we change this one to be 5%, still this

465
00:37:38,690 --> 00:37:39,870
statement is false.

466
00:37:43,050 --> 00:37:48,670
It should be smaller than alpha in order to reject

467
00:37:48,670 --> 00:37:53,770
the null hypothesis. So, B is the correct

468
00:37:53,770 --> 00:37:56,350
statement.

469
00:37:58,180 --> 00:38:02,080
Always insufficient means you don't reject null

470
00:38:02,080 --> 00:38:06,000
hypothesis. Now for D, we reject null hypothesis

471
00:38:06,000 --> 00:38:10,500
at 8%. Since this value 0.7 is smaller than alpha,

472
00:38:10,740 --> 00:38:14,700
so we reject. So this is incorrect. Now for C, be

473
00:38:14,700 --> 00:38:19,440
careful. At 5%, if this, if we change this one

474
00:38:19,440 --> 00:38:23,560
little bit, there is insufficient evidence. What

475
00:38:23,560 --> 00:38:32,320
do you think? About C. If we change part C as at 5

476
00:38:32,320 --> 00:38:36,540
% there is insufficient evidence to indicate the

477
00:38:36,540 --> 00:38:39,840
proportion of males preferring Pepsi equals.

478
00:38:44,600 --> 00:38:49,940
You cannot say equal because this one maybe yes

479
00:38:49,940 --> 00:38:53,200
maybe no you don't know the exact answer. So if we

480
00:38:53,200 --> 00:38:56,380
don't reject the null hypothesis then you don't

481
00:38:56,380 --> 00:38:58,780
have sufficient evidence in order to support each

482
00:38:58,780 --> 00:39:03,800
one. So, don't reject the zero as we mentioned

483
00:39:03,800 --> 00:39:10,660
before. Don't reject the zero does not imply

484
00:39:10,660 --> 00:39:16,840
if zero is true. It means the evidence, the data

485
00:39:16,840 --> 00:39:19,500
you have is not sufficient to support the

486
00:39:19,500 --> 00:39:25,260
alternative evidence. So, don't say equal to. So

487
00:39:25,260 --> 00:39:30,560
say don't reject rather than saying accept. So V

488
00:39:30,560 --> 00:39:31,460
is the correct answer.

489
00:39:35,940 --> 00:39:43,020
Six, seven, and eight. Construct 90% confidence

490
00:39:43,020 --> 00:39:48,380
interval, construct 95, construct 99. It's

491
00:39:48,380 --> 00:39:52,700
similar, just the critical value will be changed.

492
00:39:53,620 --> 00:39:58,380
Now my question is, which is the widest continence

493
00:39:58,380 --> 00:40:03,080
interval in this case? 99. The last one is the

494
00:40:03,080 --> 00:40:08,040
widest because here 99 is the largest continence

495
00:40:08,040 --> 00:40:11,160
limit. So that means the width of the interval is

496
00:40:11,160 --> 00:40:12,620
the largest in this case.

497
00:40:17,960 --> 00:40:23,770
For 5, 6 and 7. The question is construct either

498
00:40:23,770 --> 00:40:30,930
90%, 95% or 99% for the same question. Simple

499
00:40:30,930 --> 00:40:33,510
calculation will give the confidence interval for

500
00:40:33,510 --> 00:40:38,590
each one. My question was, which one is the widest

501
00:40:38,590 --> 00:40:43,630
confidence interval? Based on the C level, 99%

502
00:40:43,630 --> 00:40:47,350
gives the widest confidence interval comparing to

503
00:40:47,350 --> 00:41:02,100
90% and 95%. The exact answers for 5, 6 and 7, 0.5

504
00:41:02,100 --> 00:41:08,900
to 30 percent. For 95 percent, 0.2 to 32 percent.

505
00:41:10,750 --> 00:41:16,030
For 99, negative 0.3 to 0.37. So this is the

506
00:41:16,030 --> 00:41:21,970
widest. Because here we start from 5 to 30. Here

507
00:41:21,970 --> 00:41:26,030
we start from lower than 5, 2%, up to upper, for

508
00:41:26,030 --> 00:41:31,190
greater than 30, 32. Here we start from negative 3

509
00:41:31,190 --> 00:41:35,330
% up to 37. So this is the widest confidence

510
00:41:35,330 --> 00:41:41,950
interval. Number six. Number six. number six five

511
00:41:41,950 --> 00:41:44,850
six and seven are the same except we just share

512
00:41:44,850 --> 00:41:49,710
the confidence level z so here we have one nine

513
00:41:49,710 --> 00:41:54,070
six instead of one six four and two point five

514
00:41:54,070 --> 00:42:01,170
seven it's our seven six next read the table e

515
00:42:12,610 --> 00:42:19,330
Table A. Corporation randomly selects 150

516
00:42:19,330 --> 00:42:25,830
salespeople and finds that 66% who have never

517
00:42:25,830 --> 00:42:29,070
taken self-improvement course would like such a

518
00:42:29,070 --> 00:42:33,830
course. So currently, or in recent,

519
00:42:37,660 --> 00:42:46,940
It says that out of 150 sales people, find that 66

520
00:42:46,940 --> 00:42:51,000
% would

521
00:42:51,000 --> 00:42:56,720
like to take such course. The firm did a similar

522
00:42:56,720 --> 00:43:01,480
study 10 years ago. So in the past, they had the

523
00:43:01,480 --> 00:43:07,430
same study in which 60% of a random sample of 160

524
00:43:07,430 --> 00:43:12,430
salespeople wanted a self-improvement course. So

525
00:43:12,430 --> 00:43:13,710
in the past,

526
00:43:16,430 --> 00:43:25,230
into 160, and proportion is 60%. The groups are

527
00:43:25,230 --> 00:43:29,690
assumed to be independent random samples. Let Pi 1

528
00:43:29,690 --> 00:43:32,890
and Pi 2 represent the true proportion of workers

529
00:43:32,890 --> 00:43:36,030
who would like to attend a self-improvement course

530
00:43:36,030 --> 00:43:39,550
in the recent study and the past study

531
00:43:39,550 --> 00:43:44,490
respectively. So suppose Pi 1 and Pi 2. Pi 1 for

532
00:43:44,490 --> 00:43:49,470
recent study and Pi 2 for the past study. So

533
00:43:49,470 --> 00:43:53,590
that's the question. Now, question number one.

534
00:43:56,580 --> 00:44:00,220
If the firm wanted to test whether this proportion

535
00:44:00,220 --> 00:44:06,800
has changed from the previous study, which

536
00:44:06,800 --> 00:44:09,100
represents the relevant hypothesis?

537
00:44:14,160 --> 00:44:18,540
Again, the firm wanted to test whether this

538
00:44:18,540 --> 00:44:21,740
proportion has changed. From the previous study,

539
00:44:22,160 --> 00:44:25,900
which represents the relevant hypothesis in this

540
00:44:25,900 --> 00:44:26,140
case?

541
00:44:33,560 --> 00:44:40,120
Which is the correct? A is

542
00:44:40,120 --> 00:44:44,500
the correct answer. Why A is the correct answer?

543
00:44:45,000 --> 00:44:48,040
Since we are talking about proportions, so it

544
00:44:48,040 --> 00:44:51,750
should have pi. It changed, it means does not

545
00:44:51,750 --> 00:44:55,410
equal 2. So A is the correct answer. Now B is

546
00:44:55,410 --> 00:45:00,850
incorrect because why B is incorrect? Exactly

547
00:45:00,850 --> 00:45:03,770
because under H0 we have pi 1 minus pi 2 does not

548
00:45:03,770 --> 00:45:08,570
equal 0. Always equal sign appears only under the

549
00:45:08,570 --> 00:45:14,950
null hypothesis. So it's the opposite here. Now C

550
00:45:14,950 --> 00:45:21,190
and D talking about Upper tier or lower tier, but

551
00:45:21,190 --> 00:45:23,890
here we're talking about two-tiered test, so A is

552
00:45:23,890 --> 00:45:24,750
the correct answer.

553
00:45:29,490 --> 00:45:33,090
This sign null hypothesis states incorrectly,

554
00:45:34,030 --> 00:45:38,010
because under H0 should have equal sign, and for

555
00:45:38,010 --> 00:45:39,730
alternate it should be not equal to.

556
00:45:42,770 --> 00:45:43,630
Number two.

557
00:45:47,860 --> 00:45:51,840
If the firm wanted to test whether a greater

558
00:45:51,840 --> 00:45:56,680
proportion of workers would currently like to

559
00:45:56,680 --> 00:46:00,180
attend a self-improvement course than in the past,

560
00:46:00,900 --> 00:46:05,840
currently, the proportion is greater than in the

561
00:46:05,840 --> 00:46:13,680
past. Which represents the relevant hypothesis? C

562
00:46:13,680 --> 00:46:18,180
is the correct answer. Because it says a greater

563
00:46:18,180 --> 00:46:22,340
proportion of workers work currently. So by one,

564
00:46:22,420 --> 00:46:26,340
greater than by two. So C is the correct answer.

565
00:46:31,340 --> 00:46:40,140
It says that the firm wanted to test proportion of

566
00:46:40,140 --> 00:46:46,640
workers currently study

567
00:46:46,640 --> 00:46:50,320
or recent study by one represents the proportion

568
00:46:50,320 --> 00:46:55,140
of workers who would like to attend the course so

569
00:46:55,140 --> 00:46:58,080
that's by one greater than

570
00:47:01,730 --> 00:47:05,350
In the past. So it means by one is greater than by

571
00:47:05,350 --> 00:47:11,870
two. It means by one minus by two is positive. So

572
00:47:11,870 --> 00:47:14,590
the alternative is by one minus two by two is

573
00:47:14,590 --> 00:47:16,430
positive. So this one is the correct answer.

574
00:47:21,530 --> 00:47:26,910
Exactly. If if here we have what in the past

575
00:47:26,910 --> 00:47:30,430
should be it should be the correct answer.

576
00:47:34,690 --> 00:47:40,450
That's to three. Any question for going to number

577
00:47:40,450 --> 00:47:49,590
three? Any question for number two? Three. What is

578
00:47:49,590 --> 00:47:52,790
the unbiased point estimate for the difference

579
00:47:52,790 --> 00:47:54,410
between the two population proportions?

580
00:47:58,960 --> 00:48:04,360
B1 minus B2 which is straight forward calculation

581
00:48:04,360 --> 00:48:06,980
gives A the correct answer. Because the point

582
00:48:06,980 --> 00:48:13,320
estimate in this case is B1 minus B2. B1 is 66

583
00:48:13,320 --> 00:48:18,560
percent, B2 is 60 percent, so the answer is 6

584
00:48:18,560 --> 00:48:26,190
percent. So B1 minus B2 which is 6 percent. I

585
00:48:26,190 --> 00:48:32,450
think three is straightforward. Number four, what

586
00:48:32,450 --> 00:48:38,450
is or are the critical values which, when

587
00:48:38,450 --> 00:48:41,870
performing a z-test on whether population

588
00:48:41,870 --> 00:48:46,570
proportions are different at 5%. Here, yes, we are

589
00:48:46,570 --> 00:48:52,250
talking about two-tailed test, and alpha is 5%. So

590
00:48:52,250 --> 00:48:55,550
my critical values, they are two critical values,

591
00:48:55,630 --> 00:48:55,830
actually.

592
00:49:27,080 --> 00:49:31,000
What is or are the critical values when testing

593
00:49:31,000 --> 00:49:34,260
whether population proportions are different at 10

594
00:49:34,260 --> 00:49:39,240
%? The same instead here we have 10 instead of 5%.

595
00:49:40,920 --> 00:49:45,100
So A is the correct answer. So just use the table.

596
00:49:47,340 --> 00:49:51,440
Now for the previous one, we have 0 to 5, 0 to 5.

597
00:49:51,980 --> 00:49:57,740
The other one, alpha is 10%. So 0, 5 to the right,

598
00:49:57,880 --> 00:50:03,580
the same as to the left. So plus or minus 164.

599
00:50:06,700 --> 00:50:11,580
So 4 and 5 by using the z table.

600
00:50:20,560 --> 00:50:25,280
So exactly, since alpha here is 1, 0, 2, 5, so the

601
00:50:25,280 --> 00:50:27,880
area becomes smaller than, so it should be z

602
00:50:27,880 --> 00:50:32,380
greater than. So 1.106, the other one 1.645,

603
00:50:32,800 --> 00:50:38,030
number 6. What is or are? The critical value in

604
00:50:38,030 --> 00:50:42,450
testing whether the current population is higher

605
00:50:42,450 --> 00:50:50,990
than. Higher means above. Above 10. Above 10, 5%.

606
00:50:50,990 --> 00:50:55,870
So which? B.

607
00:50:58,470 --> 00:51:00,810
B is the correct. Z alpha.

608
00:51:06,700 --> 00:51:08,440
So, B is the correct answer.

609
00:51:11,200 --> 00:51:11,840
7.

610
00:51:14,740 --> 00:51:21,320
7 and 8 we should have to calculate number 1. 7

611
00:51:21,320 --> 00:51:25,880
was the estimated standard error of the difference

612
00:51:25,880 --> 00:51:29,660
between the two sample proportions. We should have

613
00:51:29,660 --> 00:51:30,740
a standard error.

614
00:51:34,620 --> 00:51:40,320
Square root, B dash 1 minus B dash multiplied by 1

615
00:51:40,320 --> 00:51:45,300
over N1 plus 1 over N2. And we have to find B dash

616
00:51:45,300 --> 00:51:49,220
here. Let's see how can we find B dash.

617
00:51:52,720 --> 00:51:59,700
B dash

618
00:51:59,700 --> 00:52:05,800
equal x1 plus x2. Now what's the value of X1?

619
00:52:10,400 --> 00:52:16,220
Exactly. Since B1 is X1 over N1. So that means X1

620
00:52:16,220 --> 00:52:26,600
is N1 times B1. So N1 is 150 times 60%. So that's

621
00:52:26,600 --> 00:52:35,980
99. And similarly, X2 N2, which is 160, times 60%

622
00:52:35,980 --> 00:52:48,420
gives 96. So your B dash is x1 plus x2 divided by

623
00:52:48,420 --> 00:52:55,200
N1 plus N2, which is 150 plus 310. So complete B

624
00:52:55,200 --> 00:52:58,760
dash versus the bold estimate of overall

625
00:52:58,760 --> 00:53:03,570
proportion So 9 and 9 plus 9 is 6.

626
00:53:06,390 --> 00:53:07,730
That's just B-.

627
00:53:13,210 --> 00:53:14,290
6 to 9.

628
00:53:17,150 --> 00:53:23,190
6 to 9. So this is not your answer. It's just B-.

629
00:53:23,770 --> 00:53:29,030
Now take this value and the square root of 6 to 9.

630
00:53:30,060 --> 00:53:36,280
times 1.629 multiplied by 1 over N1 which is 150

631
00:53:36,280 --> 00:53:44,980
plus 160. That's your standard error. B dash is

632
00:53:44,980 --> 00:53:49,080
not standard error. B dash is the bold estimate of

633
00:53:49,080 --> 00:53:53,740
overall proportion. Now simple calculation will

634
00:53:53,740 --> 00:53:59,740
give C. So C is the correct answer.

635
00:54:07,060 --> 00:54:15,600
What's the standard error of the difference

636
00:54:15,600 --> 00:54:17,600
between the two proportions given by this

637
00:54:17,600 --> 00:54:23,320
equation? Here first we have to compute P' by

638
00:54:23,320 --> 00:54:28,300
using x1 plus x2 over n1 plus n2. In this example,

639
00:54:29,420 --> 00:54:31,280
the x's are not given, but we have the

640
00:54:31,280 --> 00:54:35,010
proportions. And we know that B1 equals X1 over

641
00:54:35,010 --> 00:54:39,590
N1. So X1 equals N1 times B1. So I got 99.

642
00:54:40,290 --> 00:54:45,070
Similarly, X2 and 2 times B2 is 96. So B dash is

643
00:54:45,070 --> 00:54:51,610
629. So plug this value here, you will get 055.

644
00:54:56,170 --> 00:55:00,530
What's the value that is satisfactory to use in

645
00:55:00,530 --> 00:55:02,790
evaluating the alternative hypothesis? That there

646
00:55:02,790 --> 00:55:04,710
is a difference in the two population proportions.

647
00:55:05,350 --> 00:55:12,250
So we have to compute Z score, Z stat, which is V1

648
00:55:12,250 --> 00:55:16,870
minus V2, which is 6%, minus 0, divided by this

649
00:55:16,870 --> 00:55:24,840
amount, 0.55. Now, 0.6 over 0.5 around 1. Six over

650
00:55:24,840 --> 00:55:30,700
five, so the answer is one. So that's my Z

651
00:55:30,700 --> 00:55:31,000
statistic.

652
00:55:33,920 --> 00:55:34,900
That's number eight.

653
00:55:38,880 --> 00:55:43,100
So the answer is C is the correct answer. So

654
00:55:43,100 --> 00:55:48,140
straightforward calculations for C and D gives C

655
00:55:48,140 --> 00:55:51,260
correct answer for both seven and eight.

656
00:55:54,240 --> 00:55:59,300
So C is correct for each one. Now 9.

657
00:56:08,960 --> 00:56:15,240
In 9, the company tests to determine at 5% level

658
00:56:15,240 --> 00:56:18,680
of significance whether the population proportion

659
00:56:18,680 --> 00:56:22,850
has changed. from the previous study. As it

660
00:56:22,850 --> 00:56:24,910
changed, it means we are talking about two-tiered

661
00:56:24,910 --> 00:56:30,750
tests. Which of the following is most correct? So

662
00:56:30,750 --> 00:56:33,750
here we are talking about two-tiered tests and

663
00:56:33,750 --> 00:56:39,210
keep in mind your Z statistic is 1.093. And again,

664
00:56:39,470 --> 00:56:43,350
we are talking about two-tiered tests. So my

665
00:56:43,350 --> 00:56:44,690
rejection regions are

666
00:56:48,170 --> 00:56:53,150
Negative 196, critical values I mean. So the

667
00:56:53,150 --> 00:56:58,230
critical regions are 1.96 and above or smaller

668
00:56:58,230 --> 00:57:07,550
than minus 1.96. Now, my z statistic is 1.903. Now

669
00:57:07,550 --> 00:57:12,610
this value falls in the non-rejection region. So

670
00:57:12,610 --> 00:57:14,310
we don't reject the non-hypothesis.

671
00:57:16,900 --> 00:57:21,400
Ignore A and C, so the answer is either B or D.

672
00:57:22,260 --> 00:57:26,360
Now let's read B. Don't reject the null and

673
00:57:26,360 --> 00:57:28,820
conclude that the proportion of employees who are

674
00:57:28,820 --> 00:57:31,600
interested in self-improvement course has not

675
00:57:31,600 --> 00:57:32,100
changed.

676
00:57:37,040 --> 00:57:40,060
That's correct. Because we don't reject the null

677
00:57:40,060 --> 00:57:42,900
hypothesis. It means there is no significant

678
00:57:42,900 --> 00:57:45,760
difference. So it has not changed. Now, D, don't

679
00:57:45,760 --> 00:57:47,540
reject the null hypothesis and conclude the

680
00:57:47,540 --> 00:57:49,760
proportion of Obliques who are interested in a

681
00:57:49,760 --> 00:57:52,700
certain point has increased, which is incorrect.

682
00:57:53,640 --> 00:57:57,960
So B is the correct answer. So again, since my Z

683
00:57:57,960 --> 00:58:01,080
statistic falls in the non-rejection region, we

684
00:58:01,080 --> 00:58:04,380
don't reject the null hypothesis. So either B or D

685
00:58:04,380 --> 00:58:07,350
is the correct answer. But here we are talking

686
00:58:07,350 --> 00:58:12,190
about none or don't reject the null hypothesis.

687
00:58:12,470 --> 00:58:14,310
That means we don't have sufficient evidence

688
00:58:14,310 --> 00:58:17,610
support that there is significant change between

689
00:58:17,610 --> 00:58:20,670
the two proportions. So there is no difference. So

690
00:58:20,670 --> 00:58:23,270
it has not changed. It's the correct one. So you

691
00:58:23,270 --> 00:58:29,890
have to choose B. So B is the most correct answer.

692
00:58:30,830 --> 00:58:35,600
Now, 10, 11, and 12. Talking about constructing

693
00:58:35,600 --> 00:58:41,700
confidence interval 99, 95, and 90%. It's similar.

694
00:58:42,620 --> 00:58:46,140
And as we mentioned before, 99% will give the

695
00:58:46,140 --> 00:58:50,940
widest confidence interval. And the answers for

696
00:58:50,940 --> 00:59:04,300
these are 14, 11, 14, is negative 0.8 to 20%. For

697
00:59:04,300 --> 00:59:11,720
11, 0.5, negative 0.5 to 17. For 90%, negative 0.3

698
00:59:11,720 --> 00:59:15,420
to 0.15. So this is the widest confidence

699
00:59:15,420 --> 00:59:22,220
interval, which was for 99%. So similar as the

700
00:59:22,220 --> 00:59:26,360
previous one we had discussed. So for 99, always

701
00:59:26,360 --> 00:59:32,230
we get The widest confidence interval. Any

702
00:59:32,230 --> 00:59:37,490
question? That's all. Next time shall start

703
00:59:37,490 --> 00:59:41,350
chapter 12, Chi-square test of independence.