Question

Use a solution sheet to solve the hypothesis test problem. Go to Appendix E for the chi-square solution sheet. Round expected frequency to two decimal places College students may be interested in whether or not their majors have any effect on starting-salaries after graduation. Suppose that $300$recent graduates were surveyed as to their majors in college and their starting salaries after graduation. Table $11.45$shows the data. Conduct a test of independence.

Major	$<\backslash (50,000$	$\backslash )50,000-\backslash (68,999$	$\backslash )69,000+$
English	$5$	$20$	$5$
Engineering	$10$	$30$	$60$
Nursing	$10$	$15$	$15$
Business	$10$	$20$	$30$
Psychology	$20$	$30$	$20$

Table$11.45$

Short answer

H₀: Salaries are independent of majors

$\text{Taking}\alpha =0.05\text{, we can see that}p<\alpha \text{. This means that we reject null hypothesis.}$

$\text{We can conclude that major and salary are not independent events.}$

Step-by-step solution

Given Information

Suppose that $300$ recent graduates were surveyed as to their majors in college and their starting-salaries after graduation. Table $11.45$ shows the data. Conduct a test of independence

Table with observed values

Table with observed values:

Major	$<\$50,000$	$\$50,000-\$68,999$	$\$69,000+$	Total
English	$5$	$20$	$5$	$30$
Engineering	$10$	$30$	$60$	$100$
Nursing	$10$	$15$	$15$	$40$
Business	$10$	$20$	$30$	$60$
Psychology	$20$	$30$	$20$	$70$
Total	$55$	$115$	$130$	$300$

Table with expected values

Table with expected values:

Major	$<\$50,000$	$\$50,000-\$68,999$	$\$69,000+$	Total
English	$5.5$	$11.5$	$13.0$	$30$
Engineering	$18.33333$	$38.33333$	$43.33333$	$100$
Nursing	$7.333333$	$15.333333$	$17.333333$	$40$
Business	$11$	$23$	$26$	$60$
Psychology	$12.83333$	$26.83333$	$30.33333$	$70$
Total	$55$	$115$	$130$	$300$

Hypotheses test

We want to test these hypotheses:

H_$0$ : Salaries are independent of majors

H_$1$ : Salaries are dependent on majors

There are $5$types of Major and $3$groups of salaries. Thus, the number of degrees of freedom is

$(5-1)\cdot (3-1)=4\cdot 2=8$

$\text{We are using}{\chi }^2\text{distribution. Our test statistic is given by:}$

${\chi }^2=\sum _{i=1}^5\sum _{j=1}^3\frac{{(O{b}_{i,j}-E{x}_{i,j})}^2}{E{x}_{i,j}}$

=$33.54553$

Using the applet

$\text{Using the applet for the}{\chi }^2\text{distribution, we can find that}p\text{-value is}0.00005\text{:}$