Question

The Gallup Poll asks respondents how they would rate the honesty and ethical standards of people in different fields---very high, high, average, low, or very low. In 2005, 65% of the respondents gave medical doctors a rating of "very high or high", compared to a 67% rating for druggists. Is the difference real or chance variation? Or do you need more information to decide? If the difference is real, how would you EXPLAIN IT? Discuss briefly. You may assume that the results are based on a simple random sample of 1000 persons taken in 2005; each respondent rated clergy, medical doctors, druggists, and many other professions.

Answer 1

The percentage who rated medical doctors 'very high or high' is \(65 \%\)

The percentage who rated druggists 'very high or high' is \(67 \%\)

The results are based on independent simple random samples of 1,000 persons in each year

Let \(p_{1}=\) percentage who rated medical doctors 'very high or high

\(p_{2}=\) percentage who rated druggists 'very high or high'

The hypotheses to be tested are

\(H_{0}: p_{1}=p_{2}=0.5\)

\(H_{1}: p_{1} \neq p_{2}\)

Here sample size \(=1000\)

The sample proportion of men who rated clergy 'very high or high' in 2000 \(\hat{p}_{1}=0.65\)

The sample proportion of men who rated clergy 'very high or high' in 2005 \(\hat{p}_{2}=0.67\)

The pooled estimate of the common value of p

$$ \begin{array}{l} \hat{p}=\frac{n_{1} \hat{p}_{1}+n_{2} \hat{p}_{2}}{n_{1}+n_{2}} \\ =\frac{650+670}{1000+1000} \\ =0.66 \\ S E_{\text {pooldd }}=\sqrt{\hat{p}(1-\hat{p})\left(\frac{1}{n_{1}}+\frac{1}{n_{2}}\right)} \\ =\sqrt{0.66(1-0.66)\left(\frac{1}{1000}+\frac{1}{1000}\right)} \\ =0.0211849 \end{array} $$

The test statistic is calculated as follows:

\(Z=\frac{\hat{p}_{1}-\hat{p}_{2}}{S E_{\text {poled }}}\)

\(=\frac{0.60-0.54}{0.0211849}\)

\(\approx -0.9441\)

The P value is \(2 \times P[Z>0.9441] \approx 0.345\)

Conclusion: Since p-value is large (greater than 0.05\()\), we reject \(\mathrm{H}_{0}\) (at \(\alpha=0.05\) level of significance)

So we can conclude that the difference between the percentage who rated medical doctors 'very high or high and percentage who rated druggists very high or high' is statistically insignificant. The variation is just by chance.