Respuesta :
Answer:
95% confidence interval estimate of the difference between the proportion of women and men who trust recommendations made on LinkedIn is [0.081 , 0.279].
Step-by-step explanation:
We are given the data that shows the number of women and men who expressed that they trust recommendations made on LinkedIn in a recent survey;
Gender Women Men
Sample size 150 170
Trust Recommendations Made on LinkedIn 117 102
Firstly, the pivotal quantity for 95% confidence interval for the difference between population proportions is given by;
P.Q. = [tex]\frac{(\hat p_1-\hat p_2)-(p_1-p_2)}{\sqrt{\frac{\hat p_1(1-\hat p_1)}{n_1}+\frac{\hat p_2(1-\hat p_2)}{n_2} } }[/tex] ~ N(0,1)
where, [tex]\hat p_1[/tex] = sample proportion of women who trust recommendations made on LinkedIn = [tex]\frac{117}{150}[/tex] = 0.78
[tex]\hat p_1[/tex] = sample proportion of men who trust recommendations made on LinkedIn = [tex]\frac{102}{170}[/tex] = 0.60
[tex]n_1[/tex] = sample of women = 150
[tex]n_2[/tex] = sample of men = 170
Here for constructing 95% confidence interval we have used Two-sample z proportion statistics.
So, 95% confidence interval for the difference between population proportions, ([tex]p_1-p_2[/tex]) is ;
P(-1.96 < N(0,1) < 1.96) = 0.95 {As the critical value of z at 2.5% level
of significance are -1.96 & 1.96}
P(-1.96 < [tex]\frac{(\hat p_1-\hat p_2)-(p_1-p_2)}{\sqrt{\frac{\hat p_1(1-\hat p_1)}{n_1}+\frac{\hat p_2(1-\hat p_2)}{n_2} } }[/tex] < 1.96) = 0.95
P( [tex]-1.96 \times {\sqrt{\frac{\hat p_1(1-\hat p_1)}{n_1}+\frac{\hat p_2(1-\hat p_2)}{n_2} } }[/tex] < [tex]{(\hat p_1-\hat p_2)-(p_1-p_2)}[/tex] < [tex]1.96 \times {\sqrt{\frac{\hat p_1(1-\hat p_1)}{n_1}+\frac{\hat p_2(1-\hat p_2)}{n_2} } }[/tex] ) = 0.95
P( [tex](\hat p_1-\hat p_2)-1.96 \times {\sqrt{\frac{\hat p_1(1-\hat p_1)}{n_1}+\frac{\hat p_2(1-\hat p_2)}{n_2} } }[/tex] < ([tex]p_1-p_2[/tex]) < [tex](\hat p_1-\hat p_2)+1.96 \times {\sqrt{\frac{\hat p_1(1-\hat p_1)}{n_1}+\frac{\hat p_2(1-\hat p_2)}{n_2} } }[/tex] ) = 0.95
95% confidence interval for ([tex]p_1-p_2[/tex]) =
[[tex](\hat p_1-\hat p_2)-1.96 \times {\sqrt{\frac{\hat p_1(1-\hat p_1)}{n_1}+\frac{\hat p_2(1-\hat p_2)}{n_2} } }[/tex],[tex](\hat p_1-\hat p_2)+1.96 \times {\sqrt{\frac{\hat p_1(1-\hat p_1)}{n_1}+\frac{\hat p_2(1-\hat p_2)}{n_2} } }[/tex]]
= [[tex](0.78-0.60)-1.96 \times {\sqrt{\frac{0.78(1-0.78)}{150}+\frac{0.60(1-0.60)}{170} } }[/tex] ,[tex](0.78-0.60)+1.96 \times {\sqrt{\frac{0.78(1-0.78)}{150}+\frac{0.60(1-0.60)}{170} } }[/tex] ]
= [0.081 , 0.279]
Therefore, 95% confidence interval estimate of the difference between the proportion of women and men who trust recommendations made on LinkedIn is [0.081 , 0.279].
