Question

Will improving customer service result in higher stock prices for the companies providing the better service? "When a company's satisfaction score has improved over the prior year's results and is above the national average (75.4), studies show its shares have a good chance of outperforming the broad stock market in the long run." The following satisfaction scores of three companies for the 4th quarters of two previous years were obtained from an economic indicator. Assume that the scores are based on a poll of 50 customers from each company. Because the polling has been done for several years, the standard deviation can be assumed to equal 6 points in each case.

Company	Year 1	Year 2
Company A	73	76
Company B	74	77
Company C	77	78

(a)

For Company A, is the increase in the satisfaction score from year 1 to year 2 statistically significant? Use

α = 0.05.

(Let μ₁ = the satisfaction score for year 2 and μ₂ = the satisfaction score for year 1.)

State the hypotheses.

H₀: μ₁ − μ₂ < 0

H_a: μ₁ − μ₂ = 0

H₀: μ₁ − μ₂ > 0

H_a: μ₁ − μ₂ ≤ 0

    

H₀: μ₁ − μ₂ ≠ 0

H_a: μ₁ − μ₂ = 0

H₀: μ₁ − μ₂ = 0

H_a: μ₁ − μ₂ ≠ 0

H₀: μ₁ − μ₂ ≤ 0

H_a: μ₁ − μ₂ > 0

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

Calculate the p-value. (Round your answer to four decimal places.)

p-value =

What is your conclusion?

Do not reject H₀. There is sufficient evidence to conclude that the customer service has improved for Company A.

Reject H₀. There is insufficient evidence to conclude that the customer service has improved for Company A.    

Do not reject H₀. There is insufficient evidence to conclude that the customer service has improved for Company A.

Reject H₀. There is sufficient evidence to conclude that the customer service has improved for Company A.

(b)

Can you conclude that the year 2 score for Company A is above the national average of 75.4? Use

α = 0.05.

State the hypotheses.

H₀: μ = 75.4

H_a: μ ≠ 75.4

H₀: μ ≥ 75.4

H_a: μ < 75.4

    

H₀: μ ≠ 75.4

H_a: μ = 75.4

H₀: μ < 75.4

H_a: μ = 75.4

H₀: μ ≤ 75.4

H_a: μ > 75.4

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

Calculate the p-value. (Round your answer to four decimal places.)

p-value =

What is your conclusion?

Do not reject H₀. There is insufficient evidence to conclude that the score for Company A is above the national average.

Reject H₀. There is sufficient evidence to conclude that the score for Company A is above the national average.   

Reject H₀. There is insufficient evidence to conclude that the score for Company A is above the national average.Do not reject H₀.

There is sufficient evidence to conclude that the score for Company A is above the national average.

(c)

For Company B, is the increase from year 1 to year 2 statistically significant? Use

α = 0.05.

(Let μ₁ = the satisfaction score for year 2 and μ₂ = the satisfaction score for year 1.)

State the hypotheses.

H₀: μ₁ − μ₂ = 0

H_a: μ₁ − μ₂ ≠ 0

H₀: μ₁ − μ₂ ≠ 0

H_a: μ₁ − μ₂ = 0

    

H₀: μ₁ − μ₂ ≤ 0

H_a: μ₁ − μ₂ > 0

H₀: μ₁ − μ₂ ≥ 0

H_a: μ₁ − μ₂ < 0

H₀: μ₁ − μ₂ < 0

H_a: μ₁ − μ₂ = 0

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

Calculate the p-value. (Round your answer to four decimal places.)

p-value =

What is your conclusion?

Do not reject H₀. There is insufficient evidence to conclude that the customer service has improved for Company B.

Reject H₀. There is insufficient evidence to conclude that the customer service has improved for Company B.   

Do not reject H₀. There is sufficient evidence to conclude that the customer service has improved for Company B.

Reject H₀. There is sufficient evidence to conclude that the customer service has improved for Company B.

(d)

When conducting a hypothesis test with the values given for the standard deviation, sample size, and α, how large must the increase from year 1 to year 2 be for it to be statistically significant? (Round your answer to two decimal places.)

(e)

Use the result of part (d) to state whether the increase for Company C from year 1 to year 2 is statistically significant.

The increase from Year 1 to Year 2 for Company C  ---Select--- is is not statistically significant.

Answer 1

a) We are testing, H0: u1-u2= 0

vs H1: u1-u2>0

Test statistic: (x2-x1)/(sd/√n)

=(76-73)/6/√50 = 3.536

p-value of this one sided tn-1 test is: P(t49>3.536) = 0.0004 (from the t distribution tables)

Since the p-value of our test < significance level of 0.05, we have sufficient evidence to Reject H0 and conclude that yes scores of A have increased in year 2 from year 1 at the 5% level of significance.

So option d is correct.