Question

A company sells and installs satellite dishes and receivers for both private individuals and commercial establishments. The company accumulated a total of N = 2418 sales invoices last year. The company claims that the average sales amount per invoice was µ = 2120.55 USD. In order to verify that claim, an independent auditor randomly selects n = 242 of the invoices and determines the actual sales amounts by contacting the purchasers. When the sales amounts are averaged, the mean of the actual sales amounts for the 242 sampled invoices is x = 1843.93, while the sample standard deviation is s = 516.42..

a) Construct a 95% confidence interval for the the mean sales amount per invoice.

b) Based on this confidence interval did the company substantially overstate its average sales per invoice last year?

Answer 1

Given:

A company sells and installs satellite dishes and receivers for both private individuals and commercial establishments.

The company accumulated a total of N = 2418 sales invoices last year.

The company claims that the average sales amount per invoice was µ = 2120.55 USD.

Hypothesis test:

The null and alternative hypothesis is

H0 : = 2120.55

Ha : 2120.55

Sample size, n = 242

Sample mean, = 1843.93

sample standard deviation is s = 516.42

Significance level, = 1 - 0.95 = 0.05

At 95% confidence level the critical value of z is

z/2 = 1.96

A 95% confidence interval for the the mean sales amount per invoice :

CI = x̄ z/2 × s/√n

= 1843.93 1.96 × 516.42/√242

= 1843.93 65.066

= ( 1778.864, 1908.996)

Therefore a 95% confidence interval for the the mean sales amount per invoice is ( 1778.864, 1908.996 )

Since confidence interval does not contain the value claimed by the null hypothesis, so we reject null hypothesis, H0.

There is sufficient evidence to conclude that the company substantially overstate its average sales per invoice last year.