Question

Suppose the national mean annual salary for a school administrator is $91,000 a year. A school official took a sample of 25 school administrators in the state of Ohio to learn about salaries in that state to see if they differed from the national average.

77,600 76,000 90,700 97,200 90,700
101,800 78,700 81,300 84,200 97,600
77,500 75,700 89,400 84,300 78,700
84,600 87,700 103,400 83,800 101,300
94,700 69,200 95,400 61,500 68,800
(a) Formulate hypotheses that can be used to determine whether the population mean annual administrator salary in Ohio differs from the national mean of $91,000.
H0: μ ≤ 91,000
Ha: μ > 91,000
H0: μ > 91,000
Ha: μ ≤ 91,000
H0: μ ≥ 91,000
Ha: μ < 91,000
H0: μ < 91,000
Ha: μ ≥ 91,000
H0: μ = 91,000
Ha: μ ≠ 91,000
(b) The sample data for 25 Ohio administrators is contained in the file named Administrator.
What is the test statistic for your hypothesis test in part (a)? (Round your answer to three decimal places.)
What is the p-value for your hypothesis test in part (a)? (Round your answer to four decimal places.)
p-value =
(c) At
α = 0.05,
can your null hypothesis be rejected? What is your conclusion?
Reject H0. We can conclude that the mean annual administrator salary in Ohio differs significantly from the national mean annual salary.Do not reject H0. We cannot conclude that the mean annual administrator salary in Ohio differs significantly from the national mean annual salary. Reject H0. We cannot conclude that the mean annual administrator salary in Ohio differs significantly from the national mean annual salary.Do not reject H0. We can conclude that the mean annual administrator salary in Ohio differs significantly from the national mean annual salary.
(d) Repeat the preceding hypothesis test using the critical value approach.
State the null and alternative hypotheses.
H0: μ ≤ 91,000
Ha: μ > 91,000
H0: μ > 91,000
Ha: μ ≤ 91,000
H0: μ ≥ 91,000
Ha: μ < 91,000
H0: μ < 91,000
Ha: μ ≥ 91,000
H0: μ = 91,000
Ha: μ ≠ 91,000
Find the value of the test statistic. (Round your answer to three decimal places.)
State the critical values for the rejection rule. Use
α = 0.05.
(Round your answer to three decimal places. If the test is one-tailed, enter NONE for the unused tail.)
test statistic≤test statistic≥
State your conclusion.
Reject H0. We can conclude that the mean annual administrator salary in Ohio differs significantly from the national mean annual salary.Do not reject H0. We cannot conclude that the mean annual administrator salary in Ohio differs significantly from the national mean annual salary. Reject H0. We cannot conclude that the mean annual administrator salary in Ohio differs significantly from the national mean annual salary.Do not reject H0. We can conclude that the mean annual administrator salary in Ohio differs significantly from the national mean annual salary.

Answer 1

H0 : μ = 91000

H1 : μ ≠ 91000

Test statistic = - 2.594

Pvalue = 0.016

|Tcritical | = 2.064

Reject H0. We can conclude that the mean annual administrator salary in Ohio differs significantly from the national mean annual salary.

Reject H0. We can conclude that the mean annual administrator salary in Ohio differs significantly from the national mean annual salary.

Explanation:

The hypothesis :

H0 : μ = 91000

H1 : μ ≠ 91000

From the data given :

77600 76000 90700 97200 90700

101800 78700 81300 84200 97600

77500 75700 89400 84300 78700

84600 87700 103400 83800 101300

94700 69200 95400 61500 68800

Using calculator :

Sample mean, xbar = 85272

Sample standard deviation, s = 11039.23

Sample size, n = 25

The test statistic :

(xbar - μ) ÷ (s/√(n))

(85272 - 91000) / (11039.23/√(25)

Test statistic = - 5728 / 2207.846

Test statistic = - 2.594

The Pvalue : df = n - 1 = 25 - 1 = 24

Pvalue(-2.594, 24) = 0.0159

Decision region :

Reject H0 ; If Pvalue < α ;

α = 0.05

Using the critical value :

Decision region :

Reject H0 ; If Test statistic > |Tcritical;

Tcritical value at df = 24 ; α = 0.05 ;

|Tcritical | = 2.064

Hence,

We Reject H0 ; Since, |Test statistic| > |Tcritical|and conclude that mean salary depends differs