Question

Men versus women The National Assessment of Educational Progress (NAEP) Young Adult Literacy Assessment Survey interviewed a random sample of $1917$people $21$to $25$years old. The sample contained $840$men and $1077$women. ${}^{49}$The mean and standard deviation of scores on the NAEP's test of quantitative skills were ${\overline{x}}_1=272.40$and $s_1=59.2$for the men in the sample. For the women, the results were ${\overline{x}}_2=274.73$and $s_2=57.5$. Is the difference between the mean scores for men and women significant at the $1\%$ level? Give appropriate statistical evidence to justify your answer.

Short answer

No, there is not enough proof to help the claim that there is a difference.

Step-by-step solution

Given Information

${\overline{x}}_1=272.40$

${\overline{x}}_2=274.73$

$s_1=59.2$

$s_2=57.5$

$n_1=840$

$n_2=1077$

Explanation

The hypothesis

$H_0:{\mu }_1={\mu }_2$

$H_1:{\mu }_1\ne {\mu }_2$

The test statistic:

$t=\frac{{\overline{x}}_1-{\overline{x}}_2}{\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_1}}}$

$=\frac{272.40-274.72}{\sqrt{\frac{59.22}{840}}+\frac{57.52}{1077}}$

$=-0.866$

Explanation

The degrees of freedom:

$df=\min \left(n_1-1,n_2-1\right)$

$=\min (840-1,1077-1)$

$=839>100$

The P-value is the probability of obtaining the value of the test statistic, or a value more extreme.

localid="1650369677218" $0.30=2\times 0.15<P<2\times 0.20$

$=0.40$

localid="1650352701568" $0.30<P<0.40$

The P-value is the chance of getting the test statistic's result, or a number that is more severe.

$P>0.01=1\%\Rightarrow \text{Fail to reject}{H}_0$