Question

According to an article, the average non-mortgage debt of Americans is $23800. A simple random sample of 60 Americans was selected, and their average non mortgage debt was found to be $22500. Assume the population standard deviation of non-mortgage debt is $4600. Is there sufficient evidence to conclude that the average non-mortgage debt of Americans has decreased? Use α = 0.05.

a) State the null and alternative hypotheses in the context of the question.

b) Calculate the value of the test statistic. Check the required condition(s) to validate the test statistic.

c) Find the p-value.

d) What conclusion can we draw?

Answer 1

Solution :

Given that,

Population mean = $\mu$ = 23800

Sample mean = $\bar x$ = 22500

Population standard deviation = $\sigma$ = 4600

Sample size = n = 60

Level of significance = $\alpha$ = 0.05

a)

This is a left (One) tailed test,

The null and alternative hypothesis is,  

Ho: $\mu$ $\geq$ 23800

Ha: $\mu$ 23800

b)

The test statistics,

Z =( $\bar x$ - $\mu$ )/ ($\sigma$/$\sqrt{}$n)

= ( 22500 - 23800 ) / ( 4600 / $\sqrt{}$ 60 )

= -2.19

c)

P-value = P(Z < z )

= P(Z < -2.19 )

= 0.0143

d)

The p-value is p = 0.0143, and since p = 0.0143 < 0.05, it is concluded that the null hypothesis is rejected.

There is sufficient evidence to conclude that the average non-mortgage debt of Americans has decreased.