The rates of 6/23 and 318/811 for the age groups gives a confidence
interval for the difference snoring rate ![\underline{-0.206 < \hat{p}_1 - \hat{p}_2 < -0.056}](https://tex.z-dn.net/?f=%5Cunderline%7B-0.206%20%3C%20%5Chat%7Bp%7D_1%20-%20%5Chat%7Bp%7D_2%20%3C%20-0.056%7D)
<h3>What method is used to calculate the confidence interval for the difference in rate?</h3>
The number of younger age group, n₁ = 48 + 136 = 184
Proportion of the younger adult that snored,
= 48 ÷ 184 = ![\dfrac{6}{23}](https://tex.z-dn.net/?f=%5Cdfrac%7B6%7D%7B23%7D)
The number of older age group, n₂ = 318 + 493 = 811
Proportion of the older adult that snored,
= 318 ÷ 811 = ![\dfrac{318}{811}](https://tex.z-dn.net/?f=%5Cdfrac%7B318%7D%7B811%7D)
The confidence interval, CI, for the difference in two rate (proportion) is
given as follows;
![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}}}}](https://tex.z-dn.net/?f=C.I.%20%3D%20%5Cmathbf%7B%5Chat%7Bp%7D_1-%5Chat%7Bp%7D_2%5Cpm%20z_%7B%5Calpha%2F2%7D%5Csqrt%7B%5Cdfrac%7B%5Chat%7Bp%7D_1%5Cleft%20%281-%5Chat%7Bp%7D_1%20%20%5Cright%20%29%7D%7Bn_%7B1%7D%7D%2B%5Cdfrac%7B%5Chat%7Bp%7D_2%5Cleft%20%281-%5Chat%7Bp%7D_2%20%20%5Cright%20%29%7D%7Bn_%7B2%7D%7D%7D%7D)
The z-score at 96% confidence level is 2.05, which gives;
![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}}}](https://tex.z-dn.net/?f=CI%3B%5C%20%20%20%5Cmathbf%7B%5Cleft%28%5Cdfrac%7B6%7D%7B23%7D-%5Cdfrac%7B318%7D%7B811%7D%20%5Cright%29%5Cpm%202.05%20%5Ctimes%20%5Csqrt%7B%5Cdfrac%7B%5Cdfrac%7B6%7D%7B23%7D%20%5Cimes%20%5Cleft%20%281-%5Cdfrac%7B6%7D%7B23%7D%20%20%5Cright%20%29%7D%7B184%7D%2B%5Cdfrac%7B%5Cdfrac%7B318%7D%7B811%7D%20%5Ctimes%20%5Cleft%20%281-%5Cdfrac%7B318%7D%7B811%7D%20%20%5Cright%20%29%7D%7B811%7D%7D%7D)
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) = ![\underline{-0.206 < \hat{p}_1 - \hat{p}_2 < -0.056}](https://tex.z-dn.net/?f=%5Cunderline%7B-0.206%20%3C%20%5Chat%7Bp%7D_1%20-%20%5Chat%7Bp%7D_2%20%3C%20-0.056%7D)
Learn more about the finding the confidence interval for the mean of a sample here:
brainly.com/question/6156233