Using Confidence Intervals to Test Hypotheses When analyzing t...

Question

Using Confidence Intervals to Test Hypotheses When analyzing the last digits of telephone numbers in Port Jefferson, it is found that among 1000 randomly selected digits, 119 are zeros. If the digits are randomly selected, the proportion of zeros should be 0.1.

a. Use the critical value method with a 0.05 significance level to test the claim that the proportion of zeros equals 0.1.

a. Use the critical value method with a 0.05 significance level to test the claim that the proportion of zeros equals 0.1.

Accepted Answer

All right, hello, everyone. So this question says, a researcher analyzes the last digits of lottery ticket numbers in a city and finds that among 1500 randomly selected digits, 175 are sevens. If the digits are randomly distributed, the proportion of sevens should be 0.1. Use the critical value method with a 0.05 significance level to test the claim that the proportion of sevens equals 0.1. What is your conclusion? All right, so first, let's begin by defining our hypotheses. Now, for this problem, let's say that P is the true proportion of sevens among all of the lottery ticket numbers. So, the null hypothesis, or H knot. would state that P is equal to 0.1 as stated in the text of the question. So, by contrast, the alternative hypothesis, or H1 would be the opposite, right, that instead P is not equal to 0.1. Now, for this question, this is going to be a two-tailed test, because in this case, we're testing if the proportion is different from 0.1, in the sense that it could be either more or less than that value. So let's take a moment to find what information we do have. We have our sample size or lowercase n, which is equal to 1500. And we also have our X, which in this case is the number of sevens, equal to 175. So from here we can calculate the sample proportion, or Phat. So that's equal to 175 divided by 1500. Which gives you 0.1167. So with the sample proportion, you calculate the Z score. Using in this case, the Z test for proportions. So here, Z is going to be equal to He had Subtracted by P0 or P sub zero divided by the square root of. Peace of zero Multiplied by 1 subtracted by piece of zero. And then divided by. So Plugging in the information that we already have, Z would be equal to 0.1167. Subtracted by 0.1. And then divided by. The square root of 0.1, multiplied by 1 subtracted by 0.1. And then divided by. 1500. So evaluating this expression gives you a Z score equal to approximately 2.16. Which you would now compare to the critical value. So The critical value for a two-tailed test with a significance level alpha of 0.05, that would happen to be. Equal to positive and negative 1.96. So because we have two critical values, they correspond to the rejection region. Meaning, if the calculated Z score is outside of the rejection region, you can reject the null hypothesis. So in this context, our calculated value of 2.16 is greater than 1.96, meaning it is outside. Of the range, which means that it falls in the rejection region. Now, very quickly, I just want to clarify something I said a little bit earlier. What I meant to say was that if the calculated. Value If the calculated Z score is outside of the range of critical values, then that's the rejection region. So in this case, because it's outside of the range. You would reject the null hypothesis. So now this leads us to the final answer, which reads that there is sufficient evidence to reject the claim that the population of sevens is 0.1. One more correction because it's actually supposed to say proportions, not populations. But there you have it. And so with that being said, thank you so very much for watching and I hope you found this helpful.

Key Concepts

Confidence Intervals

Hypothesis Testing

Critical Value Method

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