Question

The market share of a company depends on several components such as the intensity of their own advertising campaign and the level of advertising its competitor does. A large company is believed to have a market share of 64%. The CEO of the company wants to test whether or not this value is still valid since its competitor has recently launched a new advertising campaign. A random sample of 500 consumers reveals that 244 of them use the company's product. Calculate a 90% confidence interval for the proportion of consumers who use the company's product. Would you feel comfortable in telling this CEO that her company has lost market share? Explain.

Answer 1

Let x be the number of consumers use the company's product

n is the total number of consumers

We are asked to find confidence interval estimate for population proportion P

We are given x = 244 and n = 500

Therefore = x/n = 244/500= 0.488

Lower bound = - E

Upper bound = + E

E is margin of error =

z is critical value follows standard normal distribution , we can find its value using z score table.

We are given confidence level = 0.90

Therefore α = 1 - 0.90 = 0.1 , 1 - (α/2) = 0.95

So we have to find z score corresponding to area 0.9500 on z score table

So z = 1.645

E =

E = 0.0368

Lower bound = - E = 0.488 - 0.0368 = 0.4512 or 45.12%

Upper bound = + E = 0.488 + 0.0368 = 0.5248 or 52.48%

Therefore 90% confidence interval estimate for the proportion of consumers use the company's product is ( 45.12% , 52.48% )

A large company is believed to have a market share of 64%.

Since confidence interval does not contain 64% , we can say the CEO that her company has lost market share.