Question

Marriage Ages. In the Statistics Norway on-line article "The Times They Are a Changing." J. Kristiansen discussed the changes in age at the time of marriage in Norway. The ages, in years, at the time of marriage for $75$Norwegian couples are presented on the WeissStats site. Use the technology of your choice to do the following.
a. Decide, at the $1\%$significance level, whether the data provide sufficient evidence to conclude that the mean age of Norwegian men at the time of marriage exceeds that of Norwegian women.
b. Find and interpret a $99\%$confidence interval for the difference between the mean ages at the time of marriage for Norwegian men and women.
c. Remove the two paired-difference (potential) outliers and repea parts (a) and (b). Compare your results to those in parts (a) and (b)

Short answer

(a) The data provide sufficient evidence to conclude that the mean age of Norwegian $m$time of marriage exceeds that of Norwegian women.

(b) There is a $99$percent chance that the age gap between Norwegian men and women at the time of marriage is between $1.191$and $5.609$.

(c) The gap in mean ages at the time of manhood for Norwegian men and women is estimated to be between $1.309$and $5.403$years old.

Step-by-step solution

Part (a) Step 1: Given information

$\alpha =1\%$

Explanation

$H_0:{\mu }_d=0$is the null hypothesis.

That is, the data does not support the conclusion that the average age of Norwegian men at the time of marriage is higher than that of Norwegian women.

Another possibility is:

$H_{\alpha }:{\mu }_d>0$.

That is enough information to conclude that the average age of Norwegian men at the time of marriage is higher than the average age of Norwegian women.

Step 2:

The value of the test statistic and the p-value using MINITAB.

Paired T for MEN - WOMEN

	N	Mean	StDev	SE Mean
Men	75	35.48	9.77	1.13
Women	75	32.08	7.03	0.81
Difference	75	3.400	7.235	0.835

99% lower bound for mean difference is$1.413$
$T$-Test of mean difference =$0(vs>0)$:$T$−Value=$4.07$
$p$−Value=$0.000$

Step: 3

Here,$P$− value $\le \alpha$, then reject the null hypothesis.

$P$− value $\le \alpha$
$0.000\le 0.01$

The P-value is $0.000$, which is less than the significance level. As a result, at a $1\%$level of significance, the null hypothesis is rejected.

As a result, the test results can be concluded to be statistically significant at the $1\%$level of significance.

Part (b) Step 1: Given information

$\alpha =1\%$

Explanation

If $P$− value $\le \alpha$, then reject the null hypothesis.

Paired T for MEN - WOMEN

	N	Mean	StDev	SE Mean
Men	75	35.48	9.77	1.13
Women	75	32.08	7.03	0.81
Difference	75	3.400	7.235	0.835

$99\%$CI for mean difference= $(1.191,5.609)$
$T$-Test of mean difference =$0(vsnot=0)$:$T$-Value =$4.07$

$p$-Value =$0.000$

Step: 3

The$99\%$confidence interval for the difference in age at marriage for Norwegian men and women based on MINITAB output is $(1.191,5.609)$.

Thus, there is a $99$percent chance that the age gap between Norwegian men and women at the time of marriage is between$1.191$and $5.609$.

Part (c) Step 1: Given information

$\alpha =1\%$

Explanation

If $P$− value ≤α, then reject the null hypothesis.

$H_0:{\mu }_d=0$is the null hypothesis.

That is, the data does not support the conclusion that the average age of Norwegian men at the time of marriage is higher than that of Norwegian women.

Another possibility is:

$H_a:{\mu }_d>0$.

That is enough information to conclude that the average age of Norwegian men at the time of marriage is higher than the average age of Norwegian women.

Here,$\alpha =0.01$.

Step 3:

From MINITAB, the value of test statistic and $p$-value.

Paired T for MEN - WOMEN

	N	Mean	StDev	SE Mean
Men	73	35.48	9.47	1.11
Women	73	32.12	7.11	0.83
Difference	73	3.356	6.611	0.774

$99\%$lower bound for mean difference is $1.515$.

$T$-Test of mean difference =$0$ (vs not =$0$): $T$-Value =$4.07$localid="1653387111825" $P$-Value =$0.000$.

Step 4:

The $P$-value is $0.000$, which is less than the significance level. As a result, at a $1\%$level of significance, the null hypothesis is rejected.

As a result, the test results can be concluded to be statistically significant at the $1\%$level of significance.

As a result, the data show that the average age of Norwegian men at the time of marriage is higher than that of Norwegian women.

Step 5:

After removing the outliers, use MINITAB to calculate the $99\%$confidence interval for the difference between the mean ages at the time of marriage for Norwegian men and women.

Paired T for MEN - WOMEN

	N	Mean	StDev	SE Mean
Men	73	35.48	9.47	1.11
Women	73	32.12	7.11	0.83
Difference	73	3.356	6.611	0.774

localid="1653387199239" $99\%$CI for mean difference is localid="1653387185882" $(1.309,5.403)$
localid="1653387208605" $T$-Test of mean difference =localid="1653387272139" $0$(vs not =localid="1653387263172" $0$):T-Value =localid="1653387216643" $4.34$localid="1653387244899" $p$-Value =localid="1653387253519" $0.000$

The confidence interval for the difference in ages at the time of marriage for Norwegian men and women is based on MINITAB output.

Thus, the difference between the mean ages of Norwegian men and women at the time of marriage is estimated to be between $(1.309and5.403)$.