Question

Handedness and Occupation Is the career someone chooses associated with being left- or right-handed? In one study \(^{20}\) a sample of Americans from a variety of professions were asked if they consider themselves left-handed, right-handed, or ambidextrous (equally skilled with the left and right hand). The results for five professions are shown in Table \(7.33 .\) (a) In this sample, what profession had the greatest proportion of left-handed people? What profession had the greatest proportion of right-handed people? (b) Test for an association between handedness and career for these five professions. State the null and alternative hypotheses, calculate the test statistic, and find the p-value. (c) What do you conclude at the \(5 \%\) significance level? What do you conclude at the \(1 \%\) significance level? $$ \begin{array}{l|rrr|r} \hline & \text { Right } & \text { Left } & \text { Ambidextrous } & \text { Total } \\ \hline \text { Psychiatrist } & 101 & 10 & 7 & 118 \\ \text { Architect } & 115 & 26 & 7 & 148 \\ \text { Orthopedic surgeon } & 121 & 5 & 6 & 132 \\ \text { Lawyer } & 83 & 16 & 6 & 105 \\ \text { Dentist } & 116 & 10 & 6 & 132 \\ \hline \text { Total } & 536 & 67 & 32 & 635 \\ \hline \end{array} $$

Short answer

The answers will vary depending on the calculated proportions and p-value. A precise answer can only be provided once a numerical calculation is conducted. However, The profession with the greatest proportion of left-handed and right-handed individuals can be identified using proportion calculations. Using the chi-square test one can assess whether there's an association between the profession and handedness. For a significance level of \(5 \%\) or \(1 \%\), if the p-value is less than the significance level, one would reject the null hypothesis, concluding that there is significant evidence to suggest an association between the professions and handedness.

Step-by-step solution

Identifying the profession with the greatest left-handed and right-handed proportion

To find the profession with the greatest proportion of left-handed and right-handed people one needs to calculate the proportion of left-handed and right-handed people for each profession. The proportion of left-handed people in a profession is determined by dividing the number of left-handed people in that profession by the total number of people in that profession. The same process can be applied to find the proportion of right-handed people. This can be executed for each profession.

Formulate null and alternative hypothesis

The null hypothesis, denoted as \(H_0\), suggests that there is no association between the variables: handedness and profession. The alternative hypothesis, denoted as \(H_a\), implies that there is an association between handedness and profession.

Compute test statistic

To investigate the hypothesis, a chi-square test statistic ought to be calculated. This test evaluates the goodness of fit of observed sample with the expected sample assuming that the null hypothesis is true. Its formula is:\[X^{2} = \sum \frac{(O_i - E_i)^2}{E_i}\]Where \(O_i\) represents observed frequencies and \(E_i\) represents expected frequencies. In the case of this exercise, expected frequencies can be calculated using formula:\[E_i = \frac{(Row \; Total)*(Column \; Total)}{(Grand \; Total)}\]for each cell. This formula is used since the chi-square test assumes that the observed frequencies follow a chi-square distribution under the null hypothesis.

Calculate p-value

The p-value is the probability of getting a result as extreme as the one observed if the null hypothesis is true. It is calculated using a chi-square distribution with \((r-1)(c-1)\) degrees of freedom where \(r\) is the number rows and \(c\) is the number of columns.

Drawing conclusion

Once the p-value is calculated, conclusions can be drawn based on the significance level. For a \(5\%\) significance level, if the p-value is less than 0.05, the null hypothesis is rejected meaning there is enough evidence to support the existence of an association between handedness and profession. For a \(1\%\) significance level, if the p-value is less than 0.01, the null hypothesis is rejected.