Question

A survey was done: Do you favor Soap XX? 1 = Yes 0 = No and the results were as follows

1	1	1	0	0	0	0	1	1	0
1	0	0	1	1	1	1	1	0	1
1	0	1	0	0	1	1	0	0	0
0	1	1	0	1	0	1	1	0	1

(The p-value is 0.55)

e) Use Hypothesis Testing (P-Value Method) for the claim that the proportion of people that FAVOR Soap XX is greater than 50% at 90% confidence level.

f) What is your conclusion for the Hypothesis Test, state it very clearly?

Answer 1

Solution

Final answers are given below. Back-up Theory and Details of Calculations follow at the end.

Part (a)

The claim that the proportion of people that FAVOR Soap XX is greater than 50% at 90% confidence level is NOT valid.

Answer 1

Part (b)

Conclusion:

Less than 50% of the population favor Soap XX. Answer 2

Back-up Theory and Details of Calculations

Let X = of people out of 30 that FAVOR Soap XX

Then, X ~ B(n, p),

where

n = 40 = sample size

and

p = 0.5 [i.e., 50%] = probability that a person favors Soap X, which is also equal to the population proportion.

Claim : Number of people that FAVOR Soap XX is greater than 50%

Hypotheses:

Null H₀ : p = p₀ = 0.5 [i.e., 50%] Vs Alternative H_A : p > 0.5 [claim]

Test Statistic:

Z = (p_nat - p₀)/√{p₀(1 - p₀)/n}

where

p_nat = sample proportion and

n = sample size.

Calculations:

Given	p > p0
p0	0.5
n	40
x	22
phat	0.5500
Zcal	0.6325
α	0.1
Zcrit	1.28156
p-value	0.2635

Distribution, Significance Level, α, Critical Value and p-value:

Under H₀, distribution of Z can be approximated by Standard Normal Distribution, provided

np₀ and np₀(1 - p₀) are both greater than 10.

So, given a level of significance of α%, Critical Value = upper α% of N(0, 1), and

p-value = P(Z > Zcal)

Using Excel Functions: Statistical NORMSINV and NORMDIST, critical value and p-value are found to be as shown in the above table.

Decision:

Since since p-value > α, H₀ is accepted.

Conclusion :

There is not enough evidence to suggest that the claim is valid.

DONE