Question

Women in the Labor Force. The Organization for Economic Cooperation and Development (OFCD) summarizes data on labor-force participation rates in O E C D in Figures. Independent simple random samples were taken of 300 U.S. women and 250 Canadian women. Of the U.S. women, 215 were found to be in the labor force; of the Canadian women. 186 were found to be in the labor force.

a. At the $5\%$significance level, do the data suggest that there is a difference between the labor-force participation rates of U.S. and Canadian women?

b. Find and interpret a $95\%$ confidence interval for the difference between the labor-force participation rates of U.S. and Canadian women.

Short answer

a) At$5\%$significance level, the data do not provide sufficient evidence to conclude that there is a difference between the labor-force participation rates of U.S. and Canadian women.

b) There is $95\%$interval for the difference between the proportions is $(-0.101675,0.0470087)$

Step-by-step solution

Part(a) Step 1: Given Information

The given values are,

$$\begin{gathered} x_1=215, \\ {n}_1=300, \\ {x}_2=186, \\ {n}_2=250, \\ \alpha =0.05 \end{gathered}$$

Part(a) Step 2: Explanation 

The null hypothesis

$H_0:p_1=p_2$

The alternative hypothesis

$H_a:p_1>p_2$

Using MINITAB output.

MINITAB output: Test and $\mathrm{CI}$two proportions

From the MINITAB, the test statistic is $-0.72$and the $p-$value is $0.473.$

$p$- value is more than the level of significance.

$p$- value $(=0.473)<a(=0.05)$

Using the rejection rule, it can be concluded that there is evidence to reject null hypothesis $H_0$at $alpha=0.005.$

Therefore, at $5\%$significance level, the data do not provide sufficient evidence to conclude that there is a difference between the labor-force participation rates of U.S. and Canadian women.

Part(b) Step 1: Given Information

The given values are,

$$\begin{gathered} x_1=215, \\ {n}_1=300, \\ {x}_2=186, \\ {n}_2=250, \\ \alpha =0.05 \end{gathered}$$

Part(b) Step 2: Explanation

Using MINITAB output.

Using the MINITAB output.

MINITAB output: Test and CI for two proportions.

From the MINITAB output, $95\%$confidence interval is $(-0.101675,0.0470087)$. Therefore, there is $95\%$confidence interval for the difference between the proportion is $(-0.101675,0.0470087)$