Question

 

A survey collected data on annual credit card charges in seven different categories of expenditures: transportation, groceries, dining out, household expenses, home furnishings, apparel, and entertainment. Using data from a sample of 42 credit card accounts, assume that each account was used to identify the annual credit card charges for groceries (population 1) and the annual credit card charges for dining out (population 2). Using the difference data, with population 1 − population 2, the sample mean difference was

d = $830,

and the sample standard deviation was

s_d = $1,129.

(a)

Formulate the null and alternative hypotheses to test for no difference between the population mean credit card charges for groceries and the population mean credit card charges for dining out.

a)H₀: μ_d ≥ 0

H_a: μ_d < 0

b)H₀: μ_d = 0

H_a: μ_d ≠ 0

     

c)H₀: μ_d ≠ 0

H_a: μ_d = 0

d)H₀: μ_d ≤ 0

H_a: μ_d > 0

e)H₀: μ_d < 0

H_a: μ_d = 0

(b)

Calculate the test statistic. (Round your answer to three decimal places.)

What is the p-value? (Round your answer to four decimal places.)

Can you conclude that the population means differ? Use a 0.05 level of significance.

The p ≤ 0.05, therefore we cannot conclude that there is a difference between the annual population mean expenditures for groceries and for dining out.The p ≤ 0.05, therefore we can conclude that there is a difference between the annual population mean expenditures for groceries and for dining out.     The p > 0.05, therefore we cannot conclude that there is a difference between the annual population mean expenditures for groceries and for dining out.The p > 0.05, therefore we can conclude that there is a difference between the annual population mean expenditures for groceries and for dining out.

(c)

What is the point estimate (in dollars) of the difference between the population means?

$

What is the 95% confidence interval estimate (in dollars) of the difference between the population means? (Round your answers to the nearest dollar.)

$  to $

Which category, groceries or dining out, has a higher population mean annual credit card charge?

The 95% confidence interval  ---Select--- is completely above is completely below contains zero. This suggests that the category with higher mean annual expenditure is  ---Select--- dining out groceries .

Answer 1

Solution:-

a)

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: u_d = 0

Alternative hypothesis: u_d ≠ 0

Note that these hypotheses constitute a two-tailed test.

Formulate an analysis plan. For this analysis, the significance level is 0.05. Using sample data, we will conduct a matched-pairs t-test of the null hypothesis.

Analyze sample data. Using sample data, we compute the standard deviation of the differences (s), the standard error (SE) of the mean difference, the degrees of freedom (DF), and the t statistic test statistic (t).

s = sqrt [ (\sum (d_i - d)² / (n - 1) ]

s = 1129

SE = s / sqrt(n)

S.E = 174.208

DF = n - 1 = 42 -1

D.F = 41

t = [ (x₁ - x₂) - D ] / SE

t = 4.76

where d_i is the observed difference for pair i, d is mean difference between sample pairs, D is the hypothesized mean difference between population pairs, and n is the number of pairs.

Since we have a two-tailed test, the P-value is the probability that a t statistic having 41 degrees of freedom is more extreme than 4.76; that is, less than - 4.76 or greater than 4.76.

Thus, the P-value = less than 0.001

Interpret results. Since the P-value (almost 0) is less than the significance level (0.05), we have to reject the null hypothesis.

Reject H0. The mean difference appears to differ from zero.

b) The point estimate (in dollars) of the difference between the population means is $830.

c) The 95% confidence interval estimate (in dollars) of the difference between the population means is C.I = ( 478.09, 1181.90).

C.I = 830 + 2.02 × 174.208

C.I = 830 + 351.901

C.I = ( 478.09, 1181.90)

The 95% confidence interval is completely above zero. This suggests that the category with higher mean annual expenditure is groceries .