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Consider the hypothesis test$H_{0} : p_{1}-p_{2} \leq 0$$H_{\mathrm{a}^{*}} p_{1}-p_{2}>0$The following results are for independent samples taken from the two populations.$\begin{array}{ll}{\text { Sample } 1} & {\text { Sample } 2} \\ {n_{1}-200} & {n_{2}-300} \\ {p_{1}-.22} & {p_{1}=.16}\end{array}$a. What is the $p$ -value?b. With $\alpha=.05,$ what is your hypothesis testing conclusion?

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a. $P=0.0446$b. There is sufficient evidence to support the claim that sample 1 has alarger proportion than sample 2 .

Intro Stats / AP Statistics

Chapter 11

Comparisons Involving Proportions and a Test of Independence

Descriptive Statistics

Confidence Intervals

The Chi-Square Distribution

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even the following information were asked to find a P value, associate it with the given information here and compare our P value to an Alfa 0.5 to test these hypotheses. So the first thing we need to find is a test statistic. And because we're working with proportions as given by this P bar here, we are going to compute a pool variance. So the formula for our P bar is the first sample size times the first sample proportion plus the second sample size times a second sample proportion over some of the first and second sample sizes. So in this situation that is equal to 200 times 0.22 plus 300 times 0.16 divided by 300 plus 200 and we get a test statistic of 0.184 So this is our P bar and now we can use this this value to compute a Z score. So our Z score is equal to the difference of our proportions. Divided by the square root of are pulled variants times one minus our pool variance times one over our first sample size plus one over our second sample size and all of that under a square root. So this is equal to 0.22 minus 0.16 over the square root of our P bar. We just found out to be 0.184 times one minus 10.184 one minus 10.184 times. Um, one over 200 plus one over 300 and all of that square rooted. And we get a Z test statistic of approximately 1.70 so we'll draw our normal curve. Z equals zero lies in the middle. Z equals 1.7 lives here and now we can check our oh, so part, eh? Heart lead. Um, so important. We were asked to find a p value, but we need to find the direction of, um of the hypothesis in order to discover our p value. So since this difference of means is greater than zero sorry, the difference of proportions is greater than zero. We're going to be doing a right tail t test Z test. Sorry. So we're looking at this value here, and this value represents the probability that Z is greater than or equal to 1.7, and that's hard to compute. So what we can do is find this area over here to the left of P of Z equals 1.7 and subtract that from one. So after that, we get a P value of 0.0 for four. You're a 0.44 So for a part, ay, our answer is P equals 0.44 And now we're asked to test this Against an Alfa of 0.5 Our P value is 0.440 point 044 Sorry, because 0.44 is less than our alphabet 0.5 We can reject no hypothesis. So does that mean there's sufficient evidence to support the claim that sample one has a larger proportion in the sample?

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