Respuesta :
Answer:
[tex] (0.68-0.56) -1.96 \sqrt{\frac{0.68(1-0.68)}{70} +\frac{0.56*(1-0.56)}{100}}= -0.0263[/tex]
[tex] (0.68-0.56) +1.96 \sqrt{\frac{0.68(1-0.68)}{70} +\frac{0.56*(1-0.56)}{100}}= 0.2663[/tex]
So then we are 95% confident that the true difference in the proportions is given by:
[tex] -0.0263 \leq p_1 -p_2 \leq 0.2663[/tex]
Step-by-step explanation:
The information given for this case is:
[tex]\hat p_1 = 0.68 , \hat p_2 = 0.56[/tex]
[tex] n_1 = 70, n_2 =100[/tex]
We want to construct a confidence interval for the difference of proportions [tex]p_1 -p_2[/tex] and for this case this confidence interval is given by:
[tex](\hat p_1 -\hat p_2) \pm z_{\alpha/2} \sqrt{\frac{\hat p_1 (1-\hat p_1)}{n_1} +\frac{\hat p_2 (1-\hat p_2)}{n_2}}[/tex]
The confidence level is 0.95 so then the significance level would be [tex]\alpha=1-0.95=0.05[/tex] and [tex]\alpha/2 =0.025[/tex] if we find the critical values for this confidence level we got:
[tex]z_{\alpha/2}= \pm 1.96[/tex]
And replacing the info we got:
[tex] (0.68-0.56) -1.96 \sqrt{\frac{0.68(1-0.68)}{70} +\frac{0.56*(1-0.56)}{100}}= -0.0263[/tex]
[tex] (0.68-0.56) +1.96 \sqrt{\frac{0.68(1-0.68)}{70} +\frac{0.56*(1-0.56)}{100}}= 0.2663[/tex]
So then we are 95% confident that the true difference in the proportions is given by:
[tex] -0.0263 \leq p_1 -p_2 \leq 0.2663[/tex]
The 95% confidence interval is (0.257,-0.017).
To understand the calculations, check below.
Critical value:
The critical value is a cut-off value that is used to mark the start of a region where the test statistic, obtained in hypothesis testing, is unlikely to fall in.
Given that,
[tex]n_1=70\\n_2=100\\\hat{p_1}=0.68\\\hat{p_1}=0.56\\SE=0.07\\\alpha=0.05[/tex]
The critical value is,
[tex]Z_{\frac{\alpha}{2} }=Z_{\frac{0.05}{2} }=1.96[/tex] (From standard normal probability table)
Therefore, the 95% confidence interval for the difference between population proportions [tex]p_1-p_2[/tex] is ,
[tex](\hat{p_1}-\hat{p_2})\pm Z_{\frac{\alpha}{2} }\times \sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1}+\frac{1}{n_2} ) }\\=(\hat{p_1}-\hat{p_2})\pm Z_{\frac{\alpha}{2} }\times SE\\=(0.68-0.56)\pm 1.96\times 0.07\\=0.12\pm 0.137\\=(0.257,-0.017)[/tex]
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