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Stress by occupation. As part of a study on the impact of
Chapter , Problem 6.124(choose chapter or problem)
As part of a study on the impact of job stress on smoking, researchers used data from the Health and Retirement Study (HRS) to collect information on 3825 ever-smoker individuals who were 50 to 64 years of age. 32 An ever-smoker is someone who was a smoker at some time in his or her life. The HRS is a biennial survey, thus providing the researchers with 17,043 person-year observations. One of the questions on the survey asked a participant how much he or she agrees or disagrees with the statement “My job involves a lot of stress.” The answers were coded as a 1 if a participant “strongly agreed” and 0 otherwise. The following table summarizes these responses by occupation.
\(\begin{array}{lcc}
\hline \text { Occupation } & \mathbf{p}^{\wedge} & \boldsymbol{n} \\
\hline \text { Professional } & 0.23 & 2447 \\
\text { Managerial } & 0.22 & 2552 \\
\text { Administrative } & 0.17 & 2309 \\
\text { Sales } & 0.15 & 1811 \\
\text { Mechanical } & 0.12 & 1979 \\
\text { Service } & 0.13 & 2592 \\
\text { Operator } & 0.12 & 2782 \\
\text { Farm } & 0.08 & 571 \\
\hline
\end{array}\)
(a) Because these responses are binary, use the formula for the standard deviation of a sample proportion (page 330) and construct 95% confidence intervals for each occupation.
(b) Summarize the results. Do there appear to be certain groups of occupations with similar stress levels?
(c) A friend questions the use of the standard deviation formula in part (a). Refer back to the binomial setting (page 322). What might your friend be concerned with?
Questions & Answers
QUESTION:
As part of a study on the impact of job stress on smoking, researchers used data from the Health and Retirement Study (HRS) to collect information on 3825 ever-smoker individuals who were 50 to 64 years of age. 32 An ever-smoker is someone who was a smoker at some time in his or her life. The HRS is a biennial survey, thus providing the researchers with 17,043 person-year observations. One of the questions on the survey asked a participant how much he or she agrees or disagrees with the statement “My job involves a lot of stress.” The answers were coded as a 1 if a participant “strongly agreed” and 0 otherwise. The following table summarizes these responses by occupation.
\(\begin{array}{lcc}
\hline \text { Occupation } & \mathbf{p}^{\wedge} & \boldsymbol{n} \\
\hline \text { Professional } & 0.23 & 2447 \\
\text { Managerial } & 0.22 & 2552 \\
\text { Administrative } & 0.17 & 2309 \\
\text { Sales } & 0.15 & 1811 \\
\text { Mechanical } & 0.12 & 1979 \\
\text { Service } & 0.13 & 2592 \\
\text { Operator } & 0.12 & 2782 \\
\text { Farm } & 0.08 & 571 \\
\hline
\end{array}\)
(a) Because these responses are binary, use the formula for the standard deviation of a sample proportion (page 330) and construct 95% confidence intervals for each occupation.
(b) Summarize the results. Do there appear to be certain groups of occupations with similar stress levels?
(c) A friend questions the use of the standard deviation formula in part (a). Refer back to the binomial setting (page 322). What might your friend be concerned with?
ANSWER:Step 1 of 3
As part of a study on the impact of job stress on smoking, researchers used data from the Health and Retirement Study (HRS) to collect information on 3825 ever-smoker individuals who were 50 to 64 years of age. An ever-smoker is someone who was a smoker at some time in his or her life. The HRS is a biennial survey, thus providing the researchers with 17,043 person-year observations. One of the questions on the survey asked a participant how much he or she agrees or disagrees with the statement “My job involves a lot of stress.” The answers were coded as a 1 if a participant “strongly agreed” and 0 otherwise. The following table summarizes these responses by occupation.
\(\begin{array}{|c|c|c|} \hline \text { Occupation } & p^{\wedge} & n \\ \hline \text { Professional } & 0.23 & 2447 \\ \hline \text { Managerial } & 0.22 & 2552 \\ \hline \text { Administrative } & 0.17 & 2309 \\ \hline \text { Sales } & 0.15 & 1811 \\ \hline \text { Mechanical } & 0.12 & 1979 \\ \hline \text { Service } & 0.13 & 2592 \\ \hline \text { Operator } & 0.12 & 2782 \\ \hline \text { Farm } & 0.08 & 571 \\ \hline \end{array}\)
To construct 95% confidence intervals for each occupation using the formula for the standard deviation of a sample proportion. The formula for the standard deviation of a sample proportion is
\(S . D=\sqrt{\frac{p^{\wedge}\left(1-p^{\wedge}\right)}{n}}\) (i)
The formula for 95% confidence interval is
\(C: p^{\wedge} \pm m\) (ii)
In formula (ii), \(m\) is the margin of error. Using formulae (i) and (ii), the confidence interval for each occupation is shown in the table given below.
\(\begin{array}{|c|c|c|c|c|c|} \hline \text { Occupation } & p^{\wedge} & n & \text { S.D } & m=1.96 \times \text { S.D } & \text { C.I } \\ \hline \text { Professional } & 0.23 & 2447 & 0.00851 & 0.01667 & {[0.2133,0.2467]} \\ \hline \text { Managerial } & 0.22 & 2552 & 0.00820 & 0.01607 & {[0.2039,0.2361]} \\ \hline \text { Administrative } & 0.17 & 2309 & 0.00782 & 0.01532 & {[0.1547,0.1853]} \\ \hline \text { Sales } & 0.15 & 1811 & 0.00839 & 0.01645 & {[0.1336,0.1664]} \\ \hline \text { Mechanical } & 0.12 & 1979 & 0.00730 & 0.01432 & {[0.1057,0.1343]} \\ \hline \text { Service } & 0.13 & 2592 & 0.00661 & 0.01295 & {[0.1171,0.1429]} \\ \hline \text { Operator } & 0.12 & 2782 & 0.00616 & 0.01208 & {[0.1079,0.1321]} \\ \hline \text { Farm } & 0.08 & 571 & 0.01135 & 0.02225 & {[0.0577,0.1023]} \\ \hline \end{array}\)