Question

Are most student government leaders extroverts? According to Myers-Briggs estimates, about 82% of college student government leaders are extroverts.† Suppose that a Myers-Briggs personality preference test was given to a random sample of 70 student government leaders attending a large national leadership conference and that 57 were found to be extroverts. Does this indicate that the population proportion of extroverts among college student government leaders is different (either way) from 82%? Use α = 0.01.

(a) What is the level of significance?


State the null and alternate hypotheses.

H₀: p = 0.82; H₁: p ≠ 0.82H₀: p = 0.82; H₁: p < 0.82    H₀: p ≠ 0.82; H₁: p = 0.82H₀: p = 0.82; H₁: p > 0.82H₀: p > 0.82; H₁: p = 0.82

(b) What sampling distribution will you use?

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

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


(c) Find the P-value of the test statistic. (Round your answer to four decimal places.)


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 = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant.At the α = 0.01 level, we reject the null hypothesis and conclude the data are not statistically significant.    At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are statistically significant.At the α = 0.01 level, we fail to 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 0.01 level to conclude that the true proportion of extroverts among college student government leaders differs from 82%.There is insufficient evidence at the 0.01 level to conclude that the true proportion of extroverts among college student government leaders differs from 82%.    

Answer 1

a)
0.01

Null Hypothesis, H0: p = 0.82
Alternative Hypothesis, Ha: p ≠ 0.82

b)
The standard normal, since np > 5 and nq > 5

Test statistic,
z = (pcap - p)/sqrt(p*(1-p)/n)
z = (0.8143 - 0.82)/sqrt(0.82*(1-0.82)/70)
z = -0.12

c)
P-value = 0.9045

The shared area in the two tails of the standard normal curve

d)
At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

e)
There is insufficient evidence at the 0.01 level to conclude that the true proportion of extroverts among college student government leaders differs from 82%.