Question

A blind taste test is conducted to determine which of two​ colas, Brand A or Brand​ B, individuals prefer. Individuals are randomly asked to drink one of the two types of cola​ first, followed by the other​ cola, and then asked to disclose the drink they prefer. Results of the taste test indicate that 58 of 100 individuals prefer Brand A. Complete parts a through c.

Conduct a hypothesis test​ (preferably using​ technology) Upper H 0​: p equals p 0 versus Upper H 1​: pnot equalsp 0 for p 0equals0.47​, 0.48​, 0.49​, ​..., 0.67​, 0.68​, 0.69 at the alphaequals0.05 level of significance. For which values of p 0 do you not reject the null​ hypothesis? What do each of the values of p 0 ​represent?

Do not reject the null hypothesis for the values of p 0 between ? and ? inclusively.
​(Type integers or decimals as​ needed.)

​(b) Construct a​ 95% confidence interval for the proportion of individuals who prefer Brand A.
The lower bound is
The upper bound is
​(Round to three decimal places as​ needed.)

​(c) Suppose you changed the level of significance in conducting the hypothesis test to alphaequals0.01. What would happen to the range of values for p 0 for which the null hypothesis is not​ rejected? Why does this make​ sense? Choose the correct answer below.
A.
The range of values would increase because the corresponding confidence interval would decrease in size.
B.
The range of values would decrease because the corresponding confidence interval would increase in size.
C.
The range of values would increase because the corresponding confidence interval would increase in size.
.D.
The range of values would decrease because the corresponding confidence interval would decrease in size.

Answer 1

a)

The following information is provided: The sample size is N = 100 , the number of favorable cases is X = 58 , and the sample proportion is , and the significance level is α=0.05

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: p = 0.5

Ha: p ≠ ​0.5

This corresponds to a two-tailed test, for which a z-test for one population proportion needs to be used.

(2) Rejection Region

Based on the information provided, the significance level is α=0.05, and the critical value for a two-tailed test is z_c = 1.96

(3) Test Statistics

The z-statistic is computed as follows:

(4) Decision about the null hypothesis

Since it is observed that |z| = 1.6 < z_c = 1.96 , it is then concluded that the null hypothesis is not rejected.

Using the P-value approach: The p-value is p = 0.1096 6, and since p = 0.1096 > 0.05 , it is concluded that the null hypothesis is not rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is not rejected. Therefore, there is not enough evidence to claim that the population proportion p is different than p_0p0​, at the α=0.05 significance level.

​(b) Construct a​ 95% confidence interval for the proportion of individuals who prefer Brand A.

We need to construct the 95% confidence interval for the population proportion. We have been provided with the following information about the number of favorable cases:

Favorable Cases X =	58
Sample Size N =	100

The sample proportion is computed as follows, based on the sample size N = 100 and the number of favorable cases X = 58

The critical value for α=0.05 is z_c = 1.96. The corresponding confidence interval is computed as shown below:

​(c) Suppose you changed the level of significance in conducting the hypothesis test to alpha equals 0.01. What would happen to the range of values for p 0 for which the null hypothesis is not​rejected? Why does this make​ sense? Choose the correct answer below.

C.
The range of values would increase because the corresponding confidence interval would increase in size.