Question

A cell phone company offers two plans to its subscribers. At the time new subscribers sign up, they are asked to provide some demographic information. The mean yearly income for a sample of 45 subscribers to Plan A is $55,400 with a standard deviation of $9,100. This distribution is positively skewed; the coefficient of skewness is not larger. For a sample of 41 subscribers to Plan B, the mean income is $57,600 with a standard deviation of $9,700.

At the 0.02 significance level, is it reasonable to conclude the mean income of those selecting Plan B is larger?

a. State the decision rule. (Negative answer should be indicated by a minus sign. Round the final answer to 3 decimal places.)

Reject H₀ if t > _____.

b. Compute the value of the test statistic. (Negative answer should be indicated by a minus sign. Round the final answer to 3 decimal places.)

Value of the test statistic _____.   

c. What is your decision regarding the null hypothesis?

Reject/Do not reject H₀. There is enough/not enough  evidence to conclude that the mean income of those selecting Plan B is larger/not larger  .

d. What is the p-value? (Round the final answer to 4 decimal places.)

_____.

Answer 1

Solution:-

a)

D.F = 45 + 41 - 2

D.F = 84

Significance level = 0.02

t_Critical = - 2.086

Reject the null hypothesis t < - 2.086.

b)

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

Null hypothesis: u_A = u_B
Alternative hypothesis: u_A < u_B

Note that these hypotheses constitute a one-tailed test.

Formulate an analysis plan. For this analysis, the significance level is 0.02. Using sample data, we will conduct a two-sample t-test of the null hypothesis.

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

SE = 2033.4946
DF = 84

b)

t = [ (x₁ - x₂) - d ] / SE

t = - 1.082

where s₁ is the standard deviation of sample 1, s₂ is the standard deviation of sample 2, n₁ is thesize of sample 1, n₂ is the size of sample 2, x₁ is the mean of sample 1, x₂ is the mean of sample 2, d is the hypothesized difference between population means, and SE is the standard error.

Interpret results. Since the t-value (- 1.082) does not lies in the rejection region, hence we cannot reject the null hypothesis.

From the above test we do not have sufficient evidence in the favor of the claim that the mean income of those selecting Plan B is larger.

c)

Do not reject H₀. There is not enough evidence to conclude that the mean income of those selecting Plan B is larger.

d)

The observed difference in sample means produced a t statistic of - 1.082

P-value = P(t < -1.082)

Use the t-calculator to determine the p-value

P-value = 0.1412

Interpret results. Since the P-value (0.1412) is greater than the significance level (0.02), we cannot reject the null hypothesis.