Question

Men versus women The National Assessment of Educational Progress (NAEP)

Young Adult Literacy Assessment Survey interviewed separate random samples of$840$

men and $1077$women aged $21$to $25$years.

The mean and standard deviation of scores on the NAEP’s test of quantitative skills were x1=$272.40$and s1=$59.2$for the men in the sample. For the women, the results were x ̄2=$274.73$and s2=$57.5$.

a. Construct and interpret a $90$% confidence interval for the difference in mean score for

male and female young adults.

b. Based only on the interval from part (a), is there convincing evidence of a difference

in mean score for male and female young adults?

Short answer

(a)$90$% confidence interval for the difference in mean score for

male and female young adults is in between $-1.645to+1.645$

(b)No,there is no convincing evidence of a difference

in mean score for male and female young adults.

Step-by-step solution

Part (a) Step 1:Given Information

We have been given that,

For men:

Number of men =$840$

Mean score of men =$272.40$

Standard Deviation of men =$59.2$

For women:

Number of women =$1077$

Mean score of women =$274.73$

Standard Deviation of women =$57.5$

$\sigma =0.10$

Part (a) Step 2:Explanation

X₁=Observed mean in first sample

X₂=Observed mean in second sample

${\mu }_1$=mean of first population

${\mu }_2$=proportion in second population

Observed value of difference in proportions=X₁-X₂

Expected value of difference in proportions=${\mu }_{1-}{\mu }_2=0$

Standard Deviation= localid="1654275753578" $\sqrt{\frac{{{\sigma }_1}^2}{n_1}+\frac{{{\sigma }_2}^2}{n_2}}$

$\left|z\right|=\frac{Observedvalue-Expectedvalue}{S\tan dardDeviation}$

localid="1654275760409" $\left|z\right|=\frac{X_1-X_2-0}{\sqrt{{\displaystyle \frac{{{\sigma }_1}^2}{n_1}+\frac{{{\sigma }_2}^2}{n_2}}}}$

$\left|z\right|=\frac{272.40-274.73-0}{2.68}$

$\left|z\right|=-0.86$

So, the calculated value =$-0.86$

The tabulated value at $90$% confidence level =$\pm 1.645$

Part (b) Step 1:Given Information

We have to find ,

whether there is a difference in mean score for male and female young adults.

Part (b) Step 2:Explanation

Null hypotheses:There is no difference in mean score for male and female young adults.

Alternative hypotheses:There is difference in mean score for male and female young adults.

Calculated value of z =$-0.86$

Tabulated value of z =$\pm 1.645$

Since,$Z_{cal}<Z_{tab}$

Null hypotheses is not rejected.

Conclusion:There is no difference in mean score for male and female young adults.