Question

8. Why do people enroll in fitness programs? A random sample of 500 people were surveyed. 335 of them reported that an important reason for enrolling was to prevent heart disease. (o) Find the margin of error. Explain it's meaning. (b) Find a 95% confidence interval for p. 9. An air pollution index (from 1 to 25 with 25 being the most serious pollution) takes into account the amount of carbon dioxide in the air. A random sample of 42 readings taken one hour before sunset had a mean index reading of 9.7 with standard deviation 3.2. Find a 90% confidence interval for the po index reading. 302 3.2

help, show work. not sure if done correctly.

Answer 1

(Q8) People enroll in fitness programs to get motivation to workout and maintain their health and keep their body in shape.

(a) Margin of error is the permissible error that is allowed for while estimation, for a given level of significance. It is well known that all samples suffer from error, margin of error is the maximum value that this error can take

Margin of error for a proportion p is given by M.E = $Z_{\alpha /2}*\sigma /\sqrt{n}$ where $\alpha = 0.05 , \sigma = \sqrt{pcap*(1-pcap)/n}$

pcap = X/n where X is people who enrolled = 335 and n is sample size = 500

Hence pcap = 335/500 = 0.67

$Z_{\alpha /2} = Z_{0.025} = 1.96$ , $\sigma = \sqrt{.67*.33/500}$ = 0.021

Hence M.E = 1.96* 0.021 / $\sqrt{500}$ = 0.00184

(b) 95% confidence interval for p is given by

$p \epsilon (pcap- M.E, pcap+ M.E)$

$p \epsilon (0.67- 0.00184, 0.67+ 0.00184)$

$p \epsilon (0.6682, 0.6718)$