Question

Q1. Sam and Kevin want to know if the percentage of American consumers using Kally has dropped from its historical level of 37% of American consumers. Sam and Kevin gather a sample of 750 consumers and ask them if they use Kally. The results of this survey are presented below.

Uses Kally 240

Do not use Kally 510

total 750

A. If there is evidence that the percentage of U.S. consumers using Kally is less than 37%, Sam and Kevin will spend $50,000 on additional market research to find out why Kally use is declining. Sam and Kevin would like to operate with 99% confidence, 1% importance.

B. Conduct a hypothesis test to find out if there is evidence that the percentage of American consumers who use Kally is less than 37%. Should Sam and Kevin spend $50,000 on additional market research? Use the class format.  

C. What are the Type I and Type II errors in your test?

Q2.  Sam and Kevin want to know if the average amount American consumers spend on Kally each week has changed from the $5.10 per week that American consumers have historically spent on Kally. Sam and Kevin put together a random sample of 20 American consumers and find out how much each consumer spends on Kally each week. Consumer spending on Kally is known to follow a normal distribution. This sample has an average spending of $5.01, and a standard deviation of $0.25.

A. If there is evidence that the average amount spent on Kally is different from the historical $5.10 per week, Sam and Kevin will spend $500,000 to change their production line to increase the flexibility of their production process.

B. Perform a hypothesis test to determine if there is evidence that the average amount spent at Kally is different from the historical $5.10 per week. Using a significance of 5%, 95% confidence level, would you advise Sam and Kevin to increase the flexibility of their production line?

C.  Let's assume that the truth is that the average amount spent at Kally is the historic $5.10 per week. How often would your testing procedure report that there is evidence that the average amount spent at Kally is different from $5.10?

Answer 1

Solution

Q1B

Let X = number of consumers in asample of 750 consumers who use Kally.

Then, X ~ B(n, p), where n = sample size and p = probability that a consumer uses Kally, which is also equal to the population proportion of consumers who use Kally.

Claim :

Percentage of American consumers who use Kally is less than 37%.

Hypotheses:

Null H₀ : p = p₀ = 0.37 [i.e., 37%]Vs Alternative H_A : p < 0.35

Test Statistic:

Z = (p_hat - p₀)/√{p₀(1 - p₀)/n}

Where

p_hat = sample proportion and

n = sample size.

Calculations:

p0	0.37
n	750
x	240
phat	0.32
Zcal	- 2.8362
α	0.01
Zcrit	- 2.3263
p-value	0.0023

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 = lower (α/2)% of N(0, 1), and

p-value = P(Z < Zcal)

Using Excel Function: Statistical NORMSINV and NORMSDIST these are found as shown in the above table.

Decision:

Since Zcal < Zcrit, or equivalently, since p-value < α, H₀ is rejected.

Conclusion :

There is enough evidence to suggest that the null hypothesis is not valid. Hence, we conclude that

Percentage of American consumers who use Kally is less than 37%. Answer 1

Q1C

Type I error is the error of rejecting a null hypothesis when it is in fact true. Since tests are designed to restrict the probability of Type I error to the set level of significance,

probability of Type I error in the present case is 0.01. Answer 2

Type II error is the error of accepting a null hypothesis when it is in fact not true. So, in the present case,

P(Type II error)

= P(accepting the null hypothesis when p < 0.37) Answer 3