Question

7) "Very satisfied customers give the XYZ-Box video game system a rating that is at least 42. Suppose that the manufacturer of the XYZ-Box wishes to use the 65 satisfaction ratings to provide evidence supporting the claim that the mean composite satisfaction rating for the XYZ Box exceeds 42. Their random sample of 65 satisfaction ratings yields a sample mean ofz"42.954. Assuminethat σ=2.64, is the XYZ-Box video game system's mean composite rating greater than 42?

Answer 1

n = 65

X_bar = 42.954

sigma² = 2.64

H0: Mean Composite Rating ≤ 42

H1: Mean Composite Rating > 42

Nonnormally distributed populations
When the sampling is done from a nonnormally distributed population, the central limit theorem is used.

The central limit theorem
Given a population of any non-normal functional form with mean, mu and variance, sigma² the sampling distribution of, X_bar computed from samples of size n from this population will have mean, mu and variance sigma²/n and will be approximately normally distributed when the sample is large (30 or higher).

i.e. X_bar ∼ N(mu, sigma/squareroot(n) ).

Test Statistics, Z = (X_bar −mu)/(sigma/squareroot(n))

Z = (42.954-42)/(2.64/squareroot(65))

= 2.917

At alpha = 5% and right-tailed test, z(alpha) = 1.645

Since Z > z(alpha), H0 is rejected, that is we have significant evidence that Mean Composite Rating > 42 and H1 is true.