Question

Short Questions: Select true answer and explain why it is true.

i) If an economist wishes to determine whether there is evidence that mean family income in a community equals $450,000

A) either a one-tail or two-tail test could be used with equivalent results.

B) a one-tail test should be utilized.

C) a two-tail test should be utilized.

D) None of the above

ii)A professor of statistics wants to test that the average amount of money a typical college student spends per day during spring break is over $70. Based upon previous research, the population standard deviation is estimated to be $17.32. The professor surveys 35 students and finds that the mean spending is $72.43. Which of the following statements is most accurate?

A) fail to reject the null hypothesis at α ≤ 0.10

B) reject the null hypothesis at α = 0.10

C) reject the null hypothesis at α = 0.05

D) reject the null hypothesis at α = 0.01

III) Find the critical value and rejection region for the type of chi-square test with sample size n and level of significance α.

Two -tailed test,

n = 25, α = 0.10

Answer 1

i)

Since the economist wishes to establish equality, any deviation from mean should be included in the alternative. Thus, the alternative should be two-tailed. Hence, option C)

ii)

iii)

The chi sq. test is used to test hypothesis regarding the variance of the normal distribution. Now, there are two alternatives for it: In case, the mean is known, and we use deviations from that to form the sample variance, then the chisq. statistic has degrees of freedom n. However, if the population mean is unknown and we use the sample mean, then the degrees of freedom is (n-1). In both the cases, the cutoffs are : th and th quantiles.

I will list down the answers for both the cases:

Case: Known mean

Then, the distribution is

The cutoffs are: and

Thus rejection region is (0,14.61)U(37.65,)

Case: Unknown mean

The distribution now is:

The cutoffs are: and

Thus the rejection region is: (0,13.85)U(36.42,)