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K1: I want to know whether snoring rates differ for those under and over 30 years old. The data


are from a random sample of 1010 U. S. Adults surveyed in the 2002 Sleep in America Poll. Of


these, 995 responded to the question abfut snoring, indicating whether or not they had snored at


least a few nights a week in the past year. In the younger age group, 48 snored and 136 did not. In


the older group, 318 snored and 493 did not. Construct a 96% confidence interval to estimate the


difference in the snoring rates of the two age groups. You may want to do your work on paper


and scan in the document. Be sure your name and id are on the paper.


Sagot :

The rates of 6/23 and 318/811 for the age groups gives a confidence

interval for the difference snoring rate [tex]\underline{-0.206 < \hat{p}_1 - \hat{p}_2 < -0.056}[/tex]

What method is used to calculate the confidence interval for the difference in rate?

The number of younger age group, n₁  = 48 + 136 = 184

Proportion of the younger adult that snored, [tex]\mathbf{\hat{p}_1}[/tex] = 48 ÷ 184 = [tex]\dfrac{6}{23}[/tex]

The number of older age group, n₂ = 318 + 493 = 811

Proportion of the older adult that snored, [tex]\mathbf{\hat{p}_2}[/tex] = 318 ÷ 811 = [tex]\dfrac{318}{811}[/tex]

The confidence interval, CI, for the difference in two rate (proportion) is

given as follows;

[tex]C.I. = \mathbf{\hat{p}_1-\hat{p}_2\pm z_{\alpha/2}\sqrt{\dfrac{\hat{p}_1\left (1-\hat{p}_1 \right )}{n_{1}}+\dfrac{\hat{p}_2\left (1-\hat{p}_2 \right )}{n_{2}}}}[/tex]

The z-score at 96% confidence level is 2.05, which gives;

[tex]CI;\ \mathbf{\left(\dfrac{6}{23}-\dfrac{318}{811} \right)\pm 2.05 \times \sqrt{\dfrac{\dfrac{6}{23} \imes \left (1-\dfrac{6}{23} \right )}{184}+\dfrac{\dfrac{318}{811} \times \left (1-\dfrac{318}{811} \right )}{811}}}[/tex]

Which gives;

Minimum value in the interval ≈ -0.206

Interval maximum value ≈ -0.056

The confidence interval for the different in the snoring rates between the younger and the older age is therefore;

CI = (-0.206, -0.056) = [tex]\underline{-0.206 < \hat{p}_1 - \hat{p}_2 < -0.056}[/tex]

Learn more about the finding the confidence interval for the mean of a sample here:

https://brainly.com/question/6156233

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