Question

In 2008, the average household debt service ratio for homeowners was 13.2. The household debt service ratio is the ratio of debt payments (such as mortgage payments) to disposable personal income. You are asked to evaluate this debt ratio for the last 10 years using a random sample of 44 Americans with a Mean =13.88 and a sample standard deviation of 3.14. Can you infer from the data that the debt service ratio has increased since 2008? Use alpha=0.05 Repot the value of the test statistics and critical value or p-valu

Answer 1

Answer:

$t=\frac{13.88-13.2}{\frac{3.14}{\sqrt{44}}}=1.436$    

$df=n-1=44-1=43$  

$p_v =P(t_{(43)}>1.436)=0.079$  

We see that the p value i higher than the significance level so then we FAIL to reject the null hypothesis and we can conclude that the true mean is not significantly higher than 13.2 *the value of 2008 ).

Step-by-step explanation:

Information given

$\bar X=13.88$ represent the sample mean

$s=3.14$ represent the sample standard deviation

$n=44$ sample size  

$\mu_o =13.2$ represent the value that we want to test

$\alpha=0.05$ represent the significance level

t would represent the statistic (variable of interest)  

$p_v$ represent the p value for the test

Hypothesis to test

We want to conduct a hypothesis in order to check if the true mean has increased from 2008 , and the system of hypothesi are:  

Null hypothesis:$\mu \leq 13.2$  

Alternative hypothesis:$\mu > 13.2$  

The statistic for this case is:

$t=\frac{\bar X-\mu_o}{\frac{s}{\sqrt{n}}}$  (1)  

Calculating the statistic

Replacing the info given we got:

$t=\frac{13.88-13.2}{\frac{3.14}{\sqrt{44}}}=1.436$    

P-value

The degrees of freedom are:

$df=n-1=44-1=43$  

Since is a right tailed test the p value is:  

$p_v =P(t_{(43)}>1.436)=0.079$  

Decision

We see that the p value i higher than the significance level so then we FAIL to reject the null hypothesis and we can conclude that the true mean is not significantly higher than 13.2 *the value of 2008 ).