Question

The Congressional Budget Office reports that 36% of federal civilian employees have a bachelor's degree or higher (The Wall Street Journal). A random sample of 117 employees in the private sector showed that 33 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 proportion single mean (a) What is the level of significance? State the null and alternate hypotheses. H0: μ = 0.36; H1: μ ≠ 0.36 H0: p = 0.36; H1: p > 0.36 H0: p = 0.36; H1: p < 0.36 H0: μ = 0.36; H1: μ < 0.36 H0: μ = 0.36; H1: μ > 0.36 H0: p = 0.36; H1: p ≠ 0.36 (b) What sampling distribution will you use? What assumptions are you making? The Student's t, since np > 5 and nq > 5. The standard normal, 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. 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 > 0.250 0.125 < P-value < 0.250 0.050 < P-value < 0.125 0.025 < P-value < 0.050 0.005 < P-value < 0.025 P-value < 0.005 Sketch the sampling distribution and show the area corresponding to the P-value. WebAssign Plot WebAssign Plot WebAssign Plot WebAssign Plot (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 bachelor or higher degrees 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 bachelor or higher degrees in the private sector is less than in the federal civilian sector.

Answer 1

We are asked to identify and test the hypothesis that the percentage of employees holding bachelor's degree in the private sector is less than in the federal civilian sector given the following values

n=117, x=33=>p=x/n=33/117=0.28, P=36%=0.36(percentage of employees holding bachelor's degree or higher),Q=1-P=1-0.36=0.64

1) We are testing single proportion here because nowhere in the question is it mentioned about the average of the sample or of the population.

a) Null hypothesis

H₀:P=0.36

i.e. the percentage of employees holding bachelor's degree is 36%

Alternate hypothesis

H₁:P<0.36(one-tailed test)

i.e., the percentage of employees holding bachelor's degree is less than 36%.

b) We are asked to identify the sampling distribution and the assumptions used for the sampling distribution along with the test statistic value

We use the normal distribution if np>5 and nq>5.

From the given values we can see that,

np=117*0.36=42.12>5 and nq=117*0.64=74.88>5

Hence the sampling distribution is normal distribution since np(42.12)>5 and nq(74.88)>5.

Hence the test statistic value is -1.80.

c) The p-value from standard normal tables is 0.04(0.0359 actually) which is less than 0.05.

Hence p-value(0.04)<0.05.

Graph is not drawn to scale The shaded region corresponds to the p-value for the given test statistic value of -1.80 whose absolute value is 1.80.

d) Since p-value(0.04) is less than the significance level(0.05), we reject the null hypothesis and we conclude that the data are statistically significant.

Hence at level, we reject the null hypothesis and conclude that the data are statistically significant.

e) There is sufficient evidence at 0.05 level to conclude that the proportion of bachelor or higher degress in the private sector is less than in the federal civilian sector(this is our alternate hypothesis statement) because rejecting the null hypothesis means accepting the alternate hypothesis.