Question

Do private practice doctors and hospital doctors have the same distribution of working hours? Suppose that a sample of $100$private practice doctors and $150$hospital doctors are selected at random and asked about the number of hours a week they work. The results are shown in Table

What is the test statistic?

Short answer

The value of test statistic is$22.50368.$

Step-by-step solution

Given information

The given data is

Explanation

From the given table

$\begin{array}{|lllllll|}\hline \text{Given Table (Observed Frequency)}& & & & & & \\ & 20-30& 30-40& 40-50& 50-60& \text{Total}& \\ \text{Private Practice}& 16& 40& 38& 6& 100& \text{Row Total}\\ \text{Hospital}& 8& 44& 59& 39& 150& \\ \text{Total}& 24& 84& 97& 45& 250& \\ & \text{Column total}& & & & & \\ \hline\end{array}$

The formula to calculate the expected frequency is

$E=\frac{(rowtotal)(columntotal)}{Grandtotal}$

$\begin{array}{|llllll|}\hline \text{Expected Frequency}& & & & & \\ & 20-30& 30-40& 40-50& 50-60& \text{Total}\\ \text{Private Practice}& \frac{24\left(100\right)}{250}=9.6& \frac{84\left(100\right)}{250}=33.6& \frac{97\left(100\right)}{250}=38.8& \frac{45\left(100\right)}{250}=18& 100\\ \text{Hospital}& \frac{24\left(150\right)}{250}=14.4& \frac{84\left(150\right)}{250}=50.4& \frac{97\left(150\right)}{250}=58.2& \frac{45\left(150\right)}{250}=27& 150\\ \text{Total}& 24& 84& 97& 45& 250\\ \hline\end{array}$

Test statistics$=\sum \frac{(O-E{)}^2}{E}$

Therefore, test statistic is

$$\begin{gathered} =(13.50221+9.001473)or(7.11111+2.03175+0.02749+13.33333) \\ =22.50368. \end{gathered}$$