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A study sampled 350 upperclassmen (Group 1) and 250 underclassmen (Group 2) at high schools around the city of Portland. The study was performed at the end of the school year and asked each if they had used steroids at any point in the last school year. Of the upperclassmen, 23 claimed to have used steroids in the last school year, and of the underclassmen, 21 claimed to have used steroids. Run a 95% confidence interval to test for a significant difference in the proportions of students who used steroids. Enter the confidence interval - round to 3 decimal places.

Respuesta :

Answer:

[tex]-0.061 < P_1 -P_2< 0.025[/tex]

Step-by-step explanation:

Give data:

[tex]n_1 = 350[/tex]

[tex]n_2 =250[/tex]

[tex]x_1 = 23[/tex]

[tex]x_2 = 21[/tex]

[tex]\hat{P} 1 = \frac{x_1}{n_1} = \frac{23}{350} = 0.066[/tex]

[tex]\hat{P} 2 = \frac{x_2}{n_2} = \frac{21}{250} = 0.084[/tex]

for 95% confidence interval

[tex]\alpha = 1 - 0.95 = 0.05 and \alpha/2 = 0.025[/tex]

[tex]z_{\alpha/2} = 1.96[/tex]  from standard z- table

confidence interval for P_1  and P_2 is

[tex]\hat{P} 1 - \hat{P} 2 \pm z_{\alpja/2} \sqrt{\frac{\hat{P} 1(1-\hat{P} 1)}{n_1} +\frac{\hat{P} 2(1-\hat{P} 2)}{n_2}} [/tex]

[tex](0.066 - 0.084) \pm 1.96 \sqrt{\frac{0.066(1-0.066)}{350} +\frac{0.084(1-0.084)}{250}}[/tex]

[tex]-0.018 \pm 0.043[/tex]

confidence interval is

[tex]-0.018 - 0.043 < P_1 -P_2<-0.018+0.043[/tex]

[tex]-0.061 < P_1 -P_2< 0.025[/tex]

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