Starting Salaries. The National Association of Colleges and Employers (NACE) conducts surveys on salary offers to college graduates by field and degree. Results are published in Salary Survey. The following table provides summary statistics for starting salaries, in thousands of dollars, to samples of bachelor's-degree graduates in six fields.

At the $1\%$significance level, do the data provide sufficient evidence to conclude that a difference exists in mean starting salaries among bachelor's-degree candidates in the six fields? Note; For the degrees of freedom in this exercise:

The given data is

The level of significance $\alpha =0.01$

Let us see the test hypothesis

Null hypothesis

$H_0$: There is no difference exist in Mean starting salaries among bachelor's degree candidates in the six fields.

Alternative hypothesis

$H_a$: There is no difference exist in Mean starting salaries among bachelor's degree candidates in the six fields.

The mean of all observations is

$\overline{x}=\frac{46(55.1)+11(44.6)+30(59.1)+11(40.6)+44(62.6)+18\left(43\right)}{46+11+30+11+44+18}$

$=54.83$

The treatment sum of squares is

$SSTR=\sum n_i{\left({\overline{x}}_i-\overline{x}\right)}^2$

$=46(55.1-54.833{)}^2+11(44.6-54.833{)}^2+30(59.1-54.833{)}^2$

$+11(40.6-54.833{)}^2+44(62.6-54.833{)}^2$$+18(43-54.833{)}^2$

$=9104.43$

The error sum of squares is

$SSE=\sum \left(n_i-1\right){\left(s_i\right)}^2$

$=(46-1)(5.6{)}^2+(11-1\left)\right(4.7{)}^2+(30-1)(4{)}^2$

$+(11-1)(5{)}^2+(44-1\left)\right(5.7{)}^2+(18-1)(4.8{)}^2$

$=4134.85$

The total sum of squares is

$SST=SSTR+SSE$

$=9104.4310+4134.85$

$=13239.2810$

The mean treatment sum of squares is

$MSTR=\frac{SSTR}{k-1}$

$=\frac{37.099}{4-1}$

$=12.36$

The mean error sum of squares is

$MSE=\frac{SSE}{n-k}$

$=\frac{4134.85}{160-6}$

$=26.85$

The $F-$static is

$F-static=\frac{MSTR}{MSE}$

$=\frac{1820.886}{26.85}$

role="math" localid="1652207470757" $=67.817$

$F-$static $(67.817)>$critical value $(3.14)$

As a result, the crucial value approach

At a $1\%$significant threshold, the null hypothesis is rejected.

As a result, the results are statistically significant at the $1\%$level.

As a result, the statistics show that there is a disparity in mean starting earnings among bachelor's degree candidates in the six fields.