Statistical Inferences on Proportions: Assumptions, Sampling Distributions, Confidence Intervals, and Hypothesis Testing

Posted on Aug 2, 2024 in Mathematics

1. (8 points) In the current chapter covering statistical inferences on proportions, all of the procedures we have discussed (sampling distributions, confidence intervals and statistical tests) have involved two assumptions. What are those two assumptions? 
2. 1.  Simple random sample
3.   2.  Large enough sample for the Central Limit Theorem to apply.
4. 2.       (4 points) Consider the three statements below. Draw a circle around any (if any) and all that are valid statistical hypotheses.

                                                             H_A: p

 (12 points) On October 24^th, 2011, VCU introduced a new dining facility labeled the Laurel and Grace Place right by most of the freshman dorms. One of the tenants is Raising Cane’s Chicken Fingers. Suppose it is known that 24% of all current VCU students have eaten food from Raising Cane’s Chicken Fingers. If a simple random sample of 60 current VCU students is selected and the proportion

 who have eaten food from Raising Cane’s Chicken Fingers determined, describe completely the sampling distribution of

.

We have a simple random sample, and since  = 60(.24) = 14.4 and  = 60(1 – .24) = 45.6 are both greater than 10, the sample size is large enough for the CLT to apply.  So  = .24;  = .0551; and since the CLT applies, the shape is normal. Hence  ~ N(.24, .0551).

 (8 points) Croutons, Salads and Wraps is another tenant of Laurel and Grace Place. Of interest is to estimate p = the proportion of all VCU students who have eaten food from Croutons, Salads and Wraps. To estimate this proportion a 95% confidence interval will be calculated and the goal is that the margin of error will be no larger than .039. What is the minimum number of current VCU students that would need to be selected to allow the calculation of a 95% confidence interval with margin of error no greater than .039? It can be assumed for this problem only that the proportion of all current U.S. college students who have eaten food from Croutons, Salads and Wraps is .08 and this number can be used for this problem. Please circle your final answer.

.08(1 – .08) = 185.89  round up to 186

        We would need a sample size of at least 186 current VCU students. 

6.   (20 points) A simple random sample of 200 current VCU students was selected and whether they have eaten food from Croutons, Salads and Wraps was recorded for each. 22 out of these 200 current VCU students have eaten food from Croutons, Salads and Wraps. If appropriate, use this information to calculate and interpret a 95% confidence interval for the proportion of all current VCU students who have eaten food from Croutons, Salads and Wraps.

= (.0666, .1534)

  We have 95% confidence that the proportion of all current VCU students who have eaten food from Croutons, Salads and Wraps is between .0666 and .1534.

_____ 7.     (4 points) Is the proportion of all current VCU students who have eaten food from Croutons, Salads and Wraps in the confidence interval computed in question 6?

The proportion of all current VCU students who have eaten food from Croutons, Salads and Wraps is not known and hence we do not know if it is in the interval or not.

_____ 8.     (4 points) In question 6 a confidence interval was computed based on a sample of 200 students. If the number of students in the sample were decreased to 160, what impact would this have on the margin of error and width of the confidence interval?

(D) Both the margin of error and the width would increase.

10.    (12 points) To test the hypotheses stated in question 9, a simple random sample of 84 IHOP Express customers was selected, and 78 of these 84 IHOP Express customers waited 10 minutes or longer for food. If appropriate, use this information to test the hypotheses stated in question 9 at the a = .10 level of significance.

The assumptions are not met – = 84(1 – .90) = 8.4, which is not  10, so we can not complete the test.

11.    (4 points) Of interest is p = the proportion of all VCU students who have eaten at all three of Raising Cane’s Chicken Fingers, IHOP Express, and Croutons, Salads and Wraps. An unsubstantiated claim is that the proportion of all VCU students who have eaten at all three restaurants is .08, and of interest is to test this claim versus the alternative that the proportion of all VCU students who have eaten at all three restaurants is less than .08. State the appropriate null and alternative hypotheses that should be tested.

                      11.  H₀:  = .08  versus  H_A:

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12.    (16 points) Consider the information and hypotheses specified in question 11. A simple random sample of 400 VCU students was selected, and 20 of these students indicated that they have eaten at all three restaurants. If appropriate, use this information to test the hypotheses stated in question 11 at the a = .05 level of significance.

 = .05 

• n = 400,

   = .05                                                                 

We have a simple random sample, and both

 = 400(.08) = 32 and

 = 400(1 – .08) = 368 are > 10.

Z =

 = –2.21

• p-value = P(Z

• 

   = .05   Since p-value ₀

• There is sufficient evidence that the proportion of all VCU students who have eaten at all three restaurants is less than .08.