Question

2. A company's market share is very sensitive to both its level of advertising and the levels of its competitors' advertising. A firm known to have a 56% market share in the past wants to test whether or not this value is still valid in view of recent advertising campaigns of its competitors and its own increased level of advertising. A random sample of 500 consumers reveals that 298 of them use the company's product. Is there evidence to conclude that the company's market share is no longer 56%? Has it increased or decreased? Make your conclusions explicit. 15 points (total) a. Test at 95% b. What is your ultimate conclusion? Explain.

Answer 1

Solution:

Part a

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

Null hypothesis: H₀: The Company’s market share is 56%.

Alternative hypothesis: H_a: The Company’s market share is not 56%.

H₀: p = 0.56 versus H_a: p ≠ 0.56

This is a two tailed test.

Test statistic formula for this test is given as below:

Z = (p̂ - p)/sqrt(pq/n)

Where, p̂ = Sample proportion, p is population proportion, q = 1 - p, and n is sample size

x = number of items of interest = 298

n = sample size = 500

p̂ = x/n = 298/500 = 0.596

p = 0.56

q = 1 - p = 0.44

Z = (p̂ - p)/sqrt(pq/n)

Z = (0.596 – 0.56)/sqrt(0.56*0.44/500)

Z = 1.6217

P-value = 0.1049

(by using z-table)

We have c = 0.95 or α = 0.05

P-value > α = 0.05

So, we do not reject the null hypothesis

Part b

There is not sufficient evidence to conclude that the company’s market share is no longer 56%. The market share is increased because test statistic value is positive.