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A large automobile insurance company selected samples of single and married male policy-holders and recorded the number who made an insurance claim over the preceding three-year period$$ \begin{array}{c}{\text { Single Policyholders }} \\ {n_{1}=400} \\ {\text { Number making claims }=76}\end{array} $$$$\begin{array}{c}{\text { Married Policyholders }} \\ {n_{2}=900} \\ {\text { Number making claims }=90}\end{array}$$a. Use $a=.05 .$ Test to determine whether the claim rates differ between single and married male policyholders.b. Provide a 95$\%$ confidence interval for the difference between the proportions for the two populations.

a. There is sufficient evidence to support the claim of a difference betweenthe population proportions.b. 0.0468 to 0.1332

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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and this problem. We're trying to see whether the claim rates claims rates differ between single and male policyholders. So are no hypothesis is going to be that they are equal. So the population proportion for our first sample is equal to population proportion for our second sample. And our alternative hypothesis is going to be that these are not equal. And we can also write this as a difference of proportions as so. And now with this, we have to come up with a test statistic because we're working with population proportions, we're going to find a Z score, and the formula for a Z score is equal to mate population proportions. The difference between the sample proportion Sorry, divided by the square root. Ah, the pool variance times one minus the pool variants times all right times one over the first sample size plus one over the second sample size. So let's find out all this information. The first thing we need to find is this difference or these, um, popular the sample proportions. So the sample proportion for the first sample is equal to um 76 divided by 400. It was approximately 0.19 And for the second Ah, sample it ISS 90 over 900 which is equal to 0.1. So with these, we have to come up with a, um, pooled variants for this part right here. Never pulled variance is going to be equal to 0.19 or is going to be equal to, um the frequency in each population divided by the sample sizes of each population. So we got a pool variance approximately point one to eight. No. With this, we can come up with this Z test statistic, so this is equal to the difference of sample proportions. So that is 0.19 minus 0.1. Divided by the square root of 0.1 to 8 are pulled variants times one minus 10.1 to 8 times one over 400 plus one over 900. And after all of this, we get a Z test statistic of approximately 4.49 So what is the sea test statistic represent? If we draw a normal curve, the Z test statistic is equal. Zero here represents, um where it is 4.49 standard deviations away from Z equals zero. And because we're looking at a, um, difference. We're seeing that we're using a two sided Z test so we not only need the area to the right of Z equals 4.49 We also need the area to the left of Z equals negative 4.49 because it is two sided. So the probability that Z is less than or equal to negative 4.49 or that Z is greater than or equal to 4.49 is equal to two times the probability that Z is less than or equal to negative 4.49 And that is a really small number. That's approximately zero. So approximately zero. And now we're going to test this against an Alfa of 0.5 and R P value is approximately zero. So because, um, zero is less than 0.5 we can reject, you know? So what does that mean? That means that our population proportions are not equal to each other. We have sufficient evidence to say that the population proportions are not equal to each other. So now we have to find out a 95% confidence interval. This party 95% confidence interval, and this is equal to the point estimate. So that is the difference in sample proportions, plus or minus the easy score of Alfa over to where Alfa is equal to one minus our confidence level of 10.95 So this is 0.5 So Z of Alfa over too times the square root of our first sample proportion times one minus our first sample proportion divided by our first sample size. Plus our second sample proportion times one minus our second sample proportion divided by our second sample size so we can plug in the values. Here we get a point 19 minus 190.1 plus or minus a Z score at, um 0.25 So Z, at zero 0.25 is equal to 1.96 times the square root of our sample proportion first table portion of 0.19 times one minus 10.19 over our first sample size of 400 plus our second sample proportion of 0.1 times one minus 10.1 over 900. And with this we get we get a confidence interval of approximately 0.468 two 0.1332 So what does this mean? We're 95% confident that the true difference of means confident. True, def of proportions. Ah, lies between these two. Values, like its allies, lies between 0.468 and 0.1332

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