Question

Individuals filing federal income tax returns prior to March 31 received an average refund of $1,051. 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 ≤ $1,051
H_a: μ > $1,051H₀: μ < $1,051
H_a: μ ≥ $1,051    H₀: μ ≥ $1,051
H_a: μ < $1,051H₀: μ > $1,051
H_a: μ ≤ $1,051H₀: μ = $1,051
H_a: μ ≠ $1,051

(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 =

At

α = 0.05,

what is your conclusion?

Reject H₀. There is sufficient evidence to conclude that the mean refund of "last minute" filers is less than $1,051.Do not reject H₀. There is sufficient evidence to conclude that the mean refund of "last minute" filers is less than $1,051.    Do not reject H₀. There is insufficient evidence to conclude that the mean refund of "last minute" filers is less or equal than $1,051.Reject H₀. There is insufficient evidence to conclude that the mean refund of "last minute" filers is less than or equal $1,051.

(d)

Repeat the preceding hypothesis test using the critical value approach.

State the null and alternative hypotheses.

H₀: μ ≤ $1,051
H_a: μ > $1,051H₀: μ < $1,051
H_a: μ ≥ $1,051    H₀: μ ≥ $1,051
H_a: μ < $1,051H₀: μ > $1,051
H_a: μ ≤ $1,051H₀: μ = $1,051
H_a: μ ≠ $1,051

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.

Reject H₀. There is sufficient evidence to conclude that the mean refund of "last minute" filers is less than $1,051.

Do not reject H₀. There is sufficient evidence to conclude that the mean refund of "last minute" filers is less than $1,051.  

Do not reject H₀. There is insufficient evidence to conclude that the mean refund of "last minute" filers is less or equal than $1,051.

Reject H₀. There is insufficient evidence to conclude that the mean refund of "last minute" filers is less than or equal $1,051.

Answer 1

(a)

Correct option:

H₀: μ ≥ $1,051
H_a: μ < $1,051

(b)

(i)

SE = /

= 1600/

= 80

Test statistic is given by:
Z = (910 - 1051)/80

= - 1.76

(ii)

Table of Area Under Standard Normal Curve gives area = 0.4608

So,

p - value = 0.5 - 0.4608 = 0.0392

(iii)

Correct option:

Reject H₀. There is sufficient evidence to conclude that the mean refund of "last minute" filers is less than $1,051.

(d)

Correct option:

H₀: μ ≥ $1,051
H_a: μ < $1,051

SE = /

= 1600/

= 80

Test statistic is given by:
Z = (910 - 1051)/80

= - 1.76

= 0.05

From Table, critical value of Z = - 1.645

Test statistic is < Critical value

So,

Correct option:

Reject H₀. There is sufficient evidence to conclude that the mean refund of "last minute" filers is less than $1,051.