Question

States that rely on sales tax for revenue to fund​ education, public​ safety, and other programs often end up with budget surpluses during economic growth periods​ (when people spend more on consumer​ goods) and budget deficits during recessions​ (when people spend less on consumer​ goods).50 small retailers in a state with a growing economy were recently sampled. The sample showed a mean increase of ​$323 in additional sales tax revenue collected per retailer compared to the previous quarter. The sample standard deviation is $37. Complete parts a through c below.

a) Find a 95​% confidence interval for the mean increase in sales tax revenue for small retailers in this state.

b) What assumptions have you made in this​ inference? Are the appropriate conditions​ satisfied? a. We have assumed that the increases in revenue are random and that the distribution of the population of increases in revenue is not biased. b. We have assumed that the increases in revenue are not multimodal and that the distribution of the population of increases in revenue does not contain any outliers. c. We have assumed that the increases in revenue are independent and that the distribution of population of increases in revenue is normal.

c)Is the randomization condition​ satisfied? a. No, because the retailers came from the same city. b. Yes, there is definitely evidence to believe that the retailers were sampled at random. c. No, there is evidence to believe that the retailers were not sampled at random. d. We don't know that the retailers were sampled at random, but it is reasonable to think that they were.

d)Is the​ 10% condition​ satisfied? a. No, 50 retailers are less than 10% of all retailers b. Yes, 50 retailers are more than 10% of all retailers. c. No, 50 retailers are more than 10% of all retailers. d. Yes, 50 retailers are less than 10% of all retailers

e)Is the nearly normal condition​ satisfied? a. Yes, the sample of 50 retailers came from a distribution that is approximately normal. b. No, the sample of 50 retailers did not come from a distribution that is approximately normal. c. We don't know, but since the sample of 50 retailers is 40 or more, it is okay to proceed.

f)Explain clearly what your interval means. a. The interval contains the true mean increase in sales tax revenue for samll retailers in this state 95% of the time. b. We are 95% confident that the interval contains the true mean icnrease in sales tax rev for small retailers in this state. c. 95% of all small retailers in this state will have an increase in sales tax rev that falls within the interval. d. 95% of the increases in sales tax rev for small retailers in this state will be contained in the interval.

Answer 1

Given the sample mean is , sample standard deviation is and sample size is .

a)The confidence interval for true mean based on the sample data is

.

So the 95% CI for mean is

b) (A) , (B), (C) are valid assumptions for a small sample size.

c) Correct choice is (D)

d) The sample size should be no more than 10% of the population size.

Correct choice is (D) Yes, 50 retailers are less than 10% of all retailers.

e)Though the problem statement does not state that the data comes from a population that follows a normal model, it can be assumed since the sample size is 40 or more.

Correct choice is (C)

f) Correct choice is (B).