Question

In a study of married women without a child, two groups (women married less than two years and women married for five years) were selected at random and each woman was asked if she eventually planned to have a child. Suppose 225 of the 300 women married less than two years planned to have children someday compared to 306 of the 450 women married for five years. Can we conclude that the proportion of wives who planned to have children is significantly higher for wives married less than two years than wives married for five years? In performing this test, what is the test statistic? Select one: cross out 0.07 o a. * 70.7080 – 0.708) 2000 0.07 cross out • b. Jo.5104 -0.510) (about) 0.07 cross out C. *0.502, 2 +1 cross out O "V300 450 0.07 d. 0.752 0.682 V300 450 0.07 e. 10.75(0.25), 0.68(0.32) V 300 450 cross out O

Answer 1

$The \ correct\ option\ is\ a)\\ \\ z = \frac{0.07}{\sqrt{0.708(1-0.708)(\frac{1}{300} + \frac{1}{450})}}\\ Given\\ \\ Sample\ sizes\ n_1=300,n_2=450\\ Sample\ proportion\ \hat p_1 =\frac{X_1}{n_1}=\frac{225}{300}=0.75\\ {\bar p_2 } =\frac{X_2}{n_2}=\frac{306}{450}=0.68\\ \\ Here \ P_1: Proportion\ of\ wives\ married\ less\ than \ two \ years\ who\ planned\ to\ have\ children\\ P_2: Proportion\ of\ wives\ married\ five \ years\ who\ planned\ to\ have\ children\\ The \ Null\ and\ alternative\ hypotheses\ are\ Ho:P_1=P_2\\ Ha:P_1 > P_2 (right-tailed\ test\ )\\ The\ test\ statistic\ is\ z\ is\ computed\ as\\ \\z = \frac{\hat p_1-\hat p_2}{\sqrt{\bar p (1-\bar p)(\frac{1}{n_1} + \frac{1}{n_2})}}\\ \\ where\ \bar p = \frac{X_1+X_2}{n_1+n_2}\ =\frac{300+450}{300+450}=0.708\\ \\ z = \frac{0.75 - 0.68}{\sqrt{0.708(1-0.708)(\frac{1}{300} + \frac{1}{450})}}\\ z = \frac{0.07}{\sqrt{0.708(1-0.708)(\frac{1}{300} + \frac{1}{450})}}\\ z=2.06550\\$$Critical\ value\ for\ \alpha=0.05\ is \ zc=1.64\\ \therefore Z>Zc \ then\ reject\ the\ null\ hypothesis\ Ho\\ we\ conclude\ that\ the\ proportion\ of\ wives\ married\ less\ than\ two\ years\ who\ planned\ to\ have\ children\ is\ significantly\\ higher\ than\ the\ proportion\ of\ wives\ married\ five\ years\\$