Question

You may need to use the appropriate appendix table or technology to answer this question.

Individuals filing federal income tax returns prior to March 31 received an average refund of $1,053. Consider the population of "last-minute" filers who mail their tax return during the last five days of the income tax period (typically April 10 to April 15).

(a)

A researcher suggests that a reason individuals wait until the last five days is that on average these individuals receive lower refunds than do early filers. Develop appropriate hypotheses such that rejection of

H0

will support the researcher's contention.

H0: μ > $1,053
Ha: μ ≤ $1,053H0: μ = $1,053
Ha: μ ≠ $1,053    H0: μ ≥ $1,053
Ha: μ < $1,053H0: μ < $1,053
Ha: μ ≥ $1,053H0: μ ≤ $1,053
Ha: μ > $1,053

(b)

For a sample of 400 individuals who filed a tax return between April 10 and 15, the sample mean refund was $910. Based on prior experience, a population standard deviation of

σ = $1,600

may be assumed.

What is the test statistic? (Round your answer to two decimal places.)

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

p-value =

(c)

At

α = 0.05,

what is your conclusion?

Do not reject H0. There is sufficient evidence to conclude that the mean refund of "last minute" filers is less than $1,053.Reject H0. There is sufficient evidence to conclude that the mean refund of "last minute" filers is less than $1,053.    Reject H0. There is insufficient evidence to conclude that the mean refund of "last minute" filers is less than or equal $1,053.Do not reject H0. There is insufficient evidence to conclude that the mean refund of "last minute" filers is less or equal than $1,053.

(d)

Repeat the preceding hypothesis test using the critical value approach.

State the null and alternative hypotheses.

H0: μ > $1,053
Ha: μ ≤ $1,053H0: μ = $1,053
Ha: μ ≠ $1,053    H0: μ ≥ $1,053
Ha: μ < $1,053H0: μ < $1,053
Ha: μ ≥ $1,053H0: μ ≤ $1,053
Ha: μ > $1,053

Find the value of the test statistic. (Round your answer to two decimal places.)

State the critical values for the rejection rule. (Use α = 0.05. Round your answer to two decimal places. If the test is one-tailed, enter NONE for the unused tail.)

test statistic≤test statistic≥

State your conclusion.

Do not reject H0. There is sufficient evidence to conclude that the mean refund of "last minute" filers is less than $1,053.Reject H0. There is sufficient evidence to conclude that the mean refund of "last minute" filers is less than $1,053.    Reject H0. There is insufficient evidence to conclude that the mean refund of "last minute" filers is less than or equal $1,053.Do not reject H0. There is insufficient evidence to conclude that the mean refund of "last minute" filers is less or equal than $1,053.

Answer 1

a) Claim : On average individuals receive lower refunds than do early filers.

H₀ : µ ≥ 1053 vs H_a : µ < 1053

b) Given : = 910 , σ = 1600 , n = 400

Population standard deviation σ is known therefore we use z statistic.

Test statistic:

z =

z = -1.79

P-value = P( z ≤ -1.79 )  

P-value = 0.0367

c) α = 0.05,

As P-value is less than 0.05, We reject H₀

Reject H₀. There is sufficient evidence to conclude that the mean refund of "last minute" filers is less than $1,053.

(d) critical value approach.

H₀ : µ ≥ 1053 vs H_a : µ < 1053

Test statistic.  z = -1.79

α = 0.05

As H_a contain < sign , this is left tail test,therefore z_0.05 = -1.64

Critical values for the rejection rule : Reject H₀ , if test statistic ≤ -1.64

Conclusion : As test statistic z = -1.79 is less than -1.64

Reject H₀. There is sufficient evidence to conclude that the mean refund of "last minute" filers is less than $1,053.