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 (75.3), 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 70 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	75	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 75.3? Use

α = 0.05.

State the hypotheses.

H₀: μ ≤ 75.3

H_a: μ > 75.3

H₀: μ ≥ 75.3

H_a: μ < 75.3

    

H₀: μ = 75.3

H_a: μ ≠ 75.3

H₀: μ ≠ 75.3

H_a: μ = 75.3

H₀: μ < 75.3

H_a: μ = 75.3

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

Answer 1

There are too many questions attached in a single set .

However we shall attempt to solve each question one by one.

Let us try with the first question .

a). Here we need to check if the increase of satisfaction score   (if any) for company A is significant ?

So our basic assumption here will be to assume that the increase is significant. This will be our null hypothesis.

This completes our detailed hypothesis testing and analysis for test of significance of company A satisfaction scores.

b). Clearly here we are to test if the second year satisfaction score for company B is greater than 75.3 or not ?

Thus the hypothesis model has to be

H₀: μ ≥ 75.3

H_a: μ < 75.3

Hence we conclude that since null hypothesis is NOT rejected ,we can say with 95% confidence that the satisfaction score of company B is significantly higher than population mean of 75.3

C). This part is similar to the hypothesis testing done for comany A in part a.

We assume that second year score is higher than first year score siginificantly.

H₀: μ₁ − μ₂ ≤ 0

H_a: μ₁ − μ₂ > 0

Thus we conclude that since null hypothesis is not rejected, we have sufficient and significant evidence to believe that second year satisfaction score is higher than first year score for company B

D). We are asked ot find the expanse of the differnce between the scores of the two years for both companies A and B

In other words, we are asked ot calculate the confidence interval at 95 % confidence level for sample sizes of 70 and standard deviation of 6.

In my analysis , we have already shown that

1). The 95% confidence interval for (μ1​−μ2​ )is −3.988<μ1​−μ2​<−0.012. for company B

or in other words

0.012<(μ2​−μ1)<3.988 .This will be limit of differnece between scores from second year to first year for company B to stay siginiifcant.

2).

The 95% confidence interval for (μ1​−μ2​ )is −4.988<μ1​−μ2​<−1.012. for company A

or in other words

1.012<(μ2​−μ1)<4.988 .

This will be limit of difference between scores from second year to first year for company A to stay signifcant.