Question

College students may be interested in whether or not their majors have any effect on starting salaries after graduation. Suppose that 295 recent graduates were surveyed as to their majors in college and their starting salaries after graduation. Below are the data. Conduct a test of independence. (Use a significance level of 0.05.)

Major	< $50,000	$50,000 - $68,999	$69,000 +
English	4	20	6
Engineering	9	29	59
Nursing	10	16	14
Business	11	19	30
Psychology	20	29	19

• Part (a)

  State the null hypothesis.

  An individual's starting salary after graduation is independent of that individual's major in college.An individual's starting salary after graduation is dependent on that individual's major in college.    

• Part (b)

  State the alternative hypothesis.

  An individual's starting salary after graduation is independent on that individual's major in college.An individual's starting salary after graduation is dependent of that individual's major in college.    

• Part (c)

  What are the degrees of freedom? (Enter an exact number as an integer, fraction, or decimal.)
  (No Response)

• Part (d)

  State the distribution to use for the test.

  χ²₄

  t₄

      

  χ²₈

  t₈

• Part (e)

  What is the test statistic? (Round your answer to two decimal places.)
  (No Response)

• Part (f)

  What is the p-value? (Round your answer to four decimal places.)
  (No Response)
  
  Explain what the p-value means for this problem.If

  H₀

  is true, then there is a chance equal to the p-value that the value of the test statistic will be equal to or greater than the calculated value.If

  H₀

  is true, then there is a chance equal to the p-value that the value of the test statistic will be equal to or less than the calculated value.    If

  H₀

  is false, then there is a chance equal to the p-value that the value of the test statistic will be equal to or less than the calculated value.If

  H₀

  is false, then there is a chance equal to the p-value that the value of the test statistic will be equal to or greater than the calculated value.

• Part (g)

  Sketch a picture of this situation. Label and scale the horizontal axis, and shade the region(s) corresponding to the p-value.

• Part (h)

  Indicate the correct decision ("reject" or "do not reject" the null hypothesis) and write the appropriate conclusion.(i) Alpha (Enter an exact number as an integer, fraction, or decimal.)
  α = (No Response)
  
  (ii) Decision:

  reject the null hypothesisdo not reject the null hypothesis    

  
  (iii) Reason for decision:

  Since α < p-value, we reject the null hypothesis.Since α > p-value, we reject the null hypothesis.    Since α > p-value, we do not reject the null hypothesis.Since α < p-value, we do not reject the null hypothesis.

  
  (iv) Conclusion:

  There is sufficient evidence to conclude that an individual's starting salary after graduation is dependent on that individual's major in college.There is not sufficient evidence to conclude that an individual's starting salary after graduation is dependent on that individual's major in college.    

Answer 1

(a)

An individual's starting salary after graduation is independent of that individual's major in college.

(b)

An individual's starting salary after graduation is dependent of that individual's major in college.    

(c)

Degree of freedom: df =( number of rows -1)*(number of columns-1) = (3-1)*(5-1)=8

(d)

χ²₈

_(e)

_{Following table shows the row total and column total:}

Major	< $50,000	$50,000 - $68,999	$69,000 +	Total
English	4	20	6	30
Engineering	9	29	59	97
Nursing	10	16	14	40
Business	11	19	30	60
Psychology	20	29	19	68
Total	54	113	128	295

_{Expected frequencies will be calculated as follows:}

_{Following table shows the expected frequencies:}

Major	< $50,000	$50,000 - $68,999	$69,000 +	Total
English	5.492	11.492	13.017	30.001
Engineering	17.756	37.156	42.088	97
Nursing	7.322	15.322	17.356	40
Business	10.983	22.983	26.034	60
Psychology	12.447	26.047	29.505	67.999
Total	54	113	128	295

_{Following table shows the calculations for chi square test statistics:}

O	E	(O-E)^2/E
4	5.492	0.405328478
9	17.756	4.317838252
10	7.322	0.979470636
11	10.983	2.63134E-05
20	12.447	4.583257733
20	11.492	6.298822137
29	37.156	1.790298633
16	15.322	0.030001566
19	22.983	0.690261889
29	26.047	0.334787461
6	13.017	3.782614197
59	42.088	6.795660141
14	17.356	0.648924637
30	26.034	0.60417746
19	29.505	3.74021437
Total		35.0016839

_{Following is the test statistics:}

(f)

The p-value is: 0.0000

Excel function used for p-value: "=CHIDIST(35.00, 8)"

If H₀ is true, then there is a chance equal to the p-value that the value of the test statistic will be equal to or greater than the calculated value.

(g)

(h)

α = 0.05

(ii)

reject the null hypothesis

(iii)

Since α > p-value, we reject the null hypothesis.   

(iv)

There is sufficient evidence to conclude that an individual's starting salary after graduation is dependent on that individual's major in college.