Question

In a report prepared by the Economic Research Department of a major bank the Department manager maintains that the average annual family income on Metropolis is $48,432. What do you conclude about the validity of the report if a random sample of 400 families shows and average income of $48,574 with a standard deviation of 2000?

Answer 1

This report is valid

Explanation:

We use this z score formula to solve for a question where a random number of samples is given:

z-score is z = (x-μ)/σ/√n

where x is the raw score

μ is the population mean

σ is the population standard deviation

n = number of samples

When σ/√n = Standard error

From the above question,

x = $48,574

μ = $48,432

σ = 2000

n = 400 families

z = $48, 574 -$48,432/(2000/√400)

= $48, 574 -$48,432/(2000/20)

= $48, 574 -$48,432/100

= 1.42

The z score is 1.42

H0 = μ = $48,432

At 0.05, we reject H0 if z < - 1.96 or > 1.96

z = 1.42

Therefore, H0 cannot be rejected.

The central limit theorem also holds because a sufficiently large amount of random samples (400) where taken from the population and replaced and this causes the mean to be randomly distributed.

Therefore, from the above z score, what we can conclude about the validity of the report is that the REPORT IS VALID because H0 cannot be rejected and the central limit theorem holds.