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 = 57 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; OR
H0: p = 0.24; H1: p < 0.24.

(b) What sampling distribution will you use?
i. The standard normal, since np < 5 and nq < 5.
ii. The Student's t, since np > 5 and nq > 5.
iii. The Student's t, since np < 5 and nq < 5.
iv. 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?
i.   At the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.
ii. At the α = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant.
iii. At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant.
iv.  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.
i. 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.
ii. 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

Population proportion p = 0.24

(A) we have to check whether the color preference of all college students is different (either way) from that of the general population. It is clear that we want to test for either way, so it is a two tailed hypothesis

Thus, null hypothesis

and alternate hypothesis

Option A is correct

(B) Population proportion is p= 0.24 and sample size n = 57

So, np = 57*0.24 = 13.68 and n(1-p) = 57*(1-0.24) = 57*0.76 = 43.32

Thus, we can say that both np and n(1-p) are greater than 5, so we will use standard normal distribution.

option D

(C) Formula for sample statistics is given as z =

where p(hat) = 10/57 and po = 0.24 and n = 57

setting the values, we get

this gives us

So, required z statistics = -1.14

P value corresponding to z value of -1.14 for two tailed hypothesis using z distribution is given as

P value = 0.2543 (check -1.1 in the left most column and its corresponding 0.04 in the top row, then select the intersecting cell)

(D) It is clear that the p value is greater than 0.05 significance level, so we failed to reject the null hypothesis as the result is not significant at 0.05 significance level. Option (iv) is correct answer

(E) Since we failed to reject the null hypothesis, thus we can say that the proportion is not different than 0.24.

So, we can conclude that there is insufficient evidence to support the statement that the proportion is different than 0.24

Option B is correct