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 42 of 100 individuals prefer Brand A. Complete parts a through c. ​(a) Conduct a hypothesis test​ (preferably using​ technology) Upper H 0​: p equals p 0 versus Upper H 1​: pnot equalsp 0 for p 0equals0.31​, 0.32​, 0.33​, ​..., 0.51​, 0.52​, 0.53 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 nothing and nothing​, inclusively. ​(Type integers or decimals as​ needed.)

Answer 1

Solution:

Given: 42 of 100 individuals prefer Brand A

Thus sample proportion =

We have to conduct a hypothesis test H₀: p=p₀ Vs H₁: p p0 for 0.31​, 0.32​, 0.33​, ​..., 0.51​, 0.52​, 0.53 at the level of significance

Part

Since , confidence level = 1 - 0.05 = 0.95 = 95%

Thus find 95% confidence interval for population proportion:

where

We need to find z_c value for c=95% confidence level.

Find Area = ( 1 + c ) / 2 = ( 1 + 0.95) /2 = 1.95 / 2 = 0.9750

Look in z table for Area = 0.9750 or its closest area and find z value.

Area = 0.9750 corresponds to 1.9 and 0.06 , thus z critical value = 1.96

That is : Z_c = 1.96

Thus

Thus

So we do not reject H0 , if population proportions are in between 0.3233 and 0.5167

0.32 < 0.3233 and 0.52 > 0.5167

So if we consider p0 = 0.32 , then we will reject H0 and if p0 = 0.52 , then also we will reject H0

but if p0 = 0.33 or p0 = 0.51 , then we do not reject H0

Thus

Do not reject the null hypothesis for the values of p₀ between 0.33 and 0.51 ​, inclusively. ​

What do each of the values of p₀ ​represent?

Each of the values of p₀ ​represent population proportion.