Conduct a goodness-of-fit test to determine if the actual college majors of graduating males fit the distribution of their expected majors.

$\begin{array}{|lll|}\hline \text{Major}& \text{Men - Expected Major}& \text{Men - Actual Major}\\ \text{Arts \& Humanities}& 11.0\%& 600\\ \text{Biological Sciences}& 6.7\%& 330\\ \text{Business}& 22.7\%& 1130\\ \text{Education}& 5.8\%& 305\\ \text{Engineering}& 15.6\%& 800\\ \text{Physical Sciences}& 3.6\%& 175\\ \text{Professional}& 9.3\%& 460\\ \text{Social Sciences}& 7.6\%& 370\\ \text{Technical}& 1.8\%& 90\\ \text{Other}& 8.2\%& 400\\ \text{Undecided}& 6.6\%& 340\\ \hline\end{array}$

A goodness-of-fit test to determine if the actual college majors of graduating males fit the distribution of their expected majors.

The number of men is $600+130+1130+305+800+175+460+370+90+400+340=5000$

The table with expected values and observed values is:

$\begin{array}{|lll|}\hline \text{Major}& \text{Men - Expected Major}& \text{Men - Actual Major}\\ \text{Arts \& Humanities}& 11.0\%& 600\\ \text{Biological Sciences}& 6.7\%& 330\\ \text{Business}& 22.7\%& 1130\\ \text{Education}& 5.8\%& 305\\ \text{Engineering}& 15.6\%& 800\\ \text{Physical Sciences}& 3.6\%& 175\\ \text{Professional}& 9.3\%& 460\\ \text{Social Sciences}& 7.6\%& 370\\ \text{Technical}& 1.8\%& 90\\ \text{Other}& 8.2\%& 400\\ \text{Undecided}& 6.6\%& 340\\ \hline\end{array}$

We want to test these hypothesis:

$H_0$: The actual college majors of graduating females fit the distribution of their expected majors.

$H_1$: The actual college majors of graduating females do not fit the distribution of their expected majors.

There are $11$different types of majors, thus the number of degrees of freedom is $11-1=10$.

We are using ${\chi }^2$distribution.

Test statistic is given by

$$\begin{gathered} {\chi }^2=\frac{(550-600{)}^2}{550}+\frac{(335-330{)}^2}{335}+\frac{(1135-1130{)}^2}{1135}++\frac{(290-305{)}^2}{290}+\frac{(780-800{)}^2}{780}+ \\ +\frac{(180-175{)}^2}{180}+\frac{(465-460{)}^2}{465}++\frac{(380-370{)}^2}{380}+\frac{(90-90{)}^2}{90}+\frac{(410-400{)}^2}{410}+\frac{(330-340{)}^2}{330} \end{gathered}$$

$=4.55+0.075+0.02+0.78+0.51+0.14+0.05+0.26+0+0.24+0.3$

localid="1648730082736" $=6.933533$

Using the applet, we get that $p$-value is $0.7317$: