Question

The U.S. Office of Personnel Management reports that 51% of federal civilian employees have a bachelor's degree or higher (OPM.gov). A random sample of 106 employees in the private sector showed that 44 have a bachelor's degree or higher. Does this indicate that the percentage of employees holding bachelor's degrees or higher in the private sector is less than in the federal civilian sector? Use α = 0.05.

What are we testing in this problem?

single meansingle proportion    

(a) What is the level of significance?


State the null and alternate hypotheses.

H₀: p = 0.51; H₁: p > 0.51H₀: p = 0.51; H₁: p < 0.51    H₀: μ = 0.51; H₁: μ > 0.51H₀: μ = 0.51; H₁: μ ≠ 0.51H₀: μ = 0.51; H₁: μ < 0.51H₀: p = 0.51; H₁: p ≠ 0.51

(b) What sampling distribution will you use? What assumptions are you making?

The Student's t, since np < 5 and nq < 5.The Student's t, since np > 5 and nq > 5.    The standard normal, since np < 5 and nq < 5.The standard normal, since np > 5 and nq > 5.

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


(c) Find (or estimate) the P-value.

P-value > 0.2500.125 < P-value < 0.250    0.050 < P-value < 0.1250.020 < P-value < 0.0500.005 < P-value < 0.020P-value < 0.005

Sketch the sampling distribution and show the area corresponding to the P-value.

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α?

At the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant.    At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.At the α = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant.

(e) Interpret your conclusion in the context of the application.

There is sufficient evidence at the 0.05 level to conclude that the proportion of employees holding bachelor's degrees or higher in the private sector is less than in the federal civilian sector.There is insufficient evidence at the 0.05 level to conclude that the proportion of employees holding bachelor's degrees or higher in the private sector is less than in the federal civilian sector.    

Answer 1

The statistic software output for this problem is:

One sample proportion summary hypothesis test: p Proportion of successes Ho p 0.51 HA p 0.51 Hypothesis test results: Std. Err Proportion Count Total Sample Prop. Z-Stat P-value 44 106 0.41509434 0.048554579-1.9546181 0.0253

Level of significance = 0.05

H₀ : p = 0.51
H_A : p < 0.51

The standard normal, since np > 5 and nq > 5.

Test statistics = -1.95

0.020 < P-value < 0.05

(d)

Option a)

(e)

Option a)