Question

What is your favorite color? A large survey of countries, including the United States, China, Russia, France, Turkey, Kenya, and others, indicated that most people prefer the color blue. In fact, about 24% of the population claim blue as their favorite color.† Suppose a random sample of n = 52 college students were surveyed and r = 10 of them said that blue is their favorite color. Does this information imply that the color preference of all college students is different (either way) from that of the general population? Use α = 0.05.

(a) What is the level of significance?


State the null and alternate hypotheses.

H₀: p = 0.24; H₁:  p ≠ 0.24

H₀: p = 0.24; H₁:  p > 0.24     

H₀: p ≠ 0.24; H₁:  p = 0.24

H₀: p = 0.24; H₁:  p < 0.24

(b) What sampling distribution will you use?

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.

The Student's t, 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 = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.

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

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

There is sufficient evidence at the 0.05 level to conclude that the true proportion of college students favoring the color blue differs from 0.24.

There is insufficient evidence at the 0.05 level to conclude that the true proportion of college students favoring the color blue differs from 0.24.

Answer 1

a)
α = 0.05
Below are the null and alternative Hypothesis,
Null Hypothesis, H0: p = 0.24
Alternative Hypothesis, Ha: p ≠ 0.24

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

Test statistic,
z = (pcap - p)/sqrt(p*(1-p)/n)
z = (0.1923 - 0.24)/sqrt(0.24*(1-0.24)/52)
z = -0.81

c)
P-value Approach
P-value = 0.4179

shaded area in both tails

d)
As P-value >= 0.05, fail to reject null hypothesis.
At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

e)
There is insufficient evidence at the 0.05 level to conclude that the true proportion of college students favoring the color blue differs from 0.24.