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 54 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 0equals0.43​, 0.44​, 0.45​, ​..., 0.63​, 0.64​, 0.65 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.

Answer 1

Solution:

Here, we have to use z test for population proportion.

H₀: p = p₀ versus H₁: p ≠ p₀

We are given p₀ = 0.43, 0.44, 0.45, ..., 0.63, 0.64, 0.65

We are given n = 100, x = 54

Sample proportion = p̂ = x/n = 54/100 = 0.54

Let p₀ = 0.43

H₀: p = 0.43 versus H₁: p ≠ 0.43

We are given level of significance = α = 0.05

Test statistic formula is given as below:

Z = (p̂ - p)/sqrt(pq/n)

Where, q = 1 – p = 1 - .43 = .57

Z = (.54 - .43)/sqrt(.43*.57/100)

Z = 2.2219

P-value = 0.0263

(by using z-table or excel)

P-value < α = 0.05

So, we reject the null hypothesis

Now, we have to perform same test for p₀ = 0.43, 0.44, 0.45, ..., 0.63, 0.64, 0.65.

So, by using excel, test statistic z values and p-values are given as below:

Excel command for p-value is given as below:

=2*(1-NORMSDIST(ABS(Z)))

p0	Z	P-value	Reject/Do not reject H0
0.43	2.221882	0.0263	Reject
0.44	2.014557	0.0440	Reject
0.45	1.809068	0.0704	Do not reject
0.46	1.605145	0.1085	Do not reject
0.47	1.402527	0.1608	Do not reject
0.48	1.200961	0.2298	Do not reject
0.49	1.0002	0.3172	Do not reject
0.5	0.8	0.4237	Do not reject
0.51	0.60012	0.5484	Do not reject
0.52	0.40032	0.6889	Do not reject
0.53	0.200361	0.8412	Do not reject
0.54	0	1.0000	Do not reject
0.55	-0.20101	0.8407	Do not reject
0.56	-0.40291	0.6870	Do not reject
0.57	-0.60597	0.5445	Do not reject
0.58	-0.81044	0.4177	Do not reject
0.59	-1.0166	0.3093	Do not reject
0.6	-1.22474	0.2207	Do not reject
0.61	-1.43516	0.1512	Do not reject
0.62	-1.64817	0.0993	Do not reject
0.63	-1.86411	0.0623	Do not reject
0.64	-2.08333	0.0372	Reject
0.65	-2.30623	0.0211	Reject

Do not reject the null hypothesis for the values of p₀ between 0.45 and 0.63, inclusively.