Question

Q2) Chi-Square Test for Goodness of Fit (10 points)

A high school physics teacher wondered if his students in the senior class this year will be more likely to go into STEM (Science, Technology, Engineering, Mathematics) majors than Social Science and Liberal Arts majors. He asked each of his 50 students about their first choice for major on their college applications and conducted a Chi-square test for goodness of fit with an alpha level of .05 to see if the number of students choosing each category differs significantly.

• 20 – STEM
• 20 – Social Sciences
• 10 – Liberal Arts

a. What is the variable in this test? What type of variable is it (nominal, ordinal, or continuous)? (2 point total: 1 for each question)

b. State the null and alternative hypotheses in words (2 points total: 1 for each hypothesis)

c. Calculate X² statistic (3 points total: give 2 points if process is correct but final answer is wrong)

Answer 1

a.

The variable in this test is first choice for major on students college applications. The type of variable is nominal as it is a categorical variable with no specific order defined.

b.

Null Hypothesis H0: The number of students choosing each category of majors are equal.

Alternative Hypothesis Ha: The number of students choosing each category of majors are not equal.

c.

Expected number of students to apply for each of three majors = (1/3) * 50 = 16.66667

X² statistic = $\sum (O_i-E_i)^2 / E_i$ where Oi and Ei are the observed and expected counts for each majors.

X² statistic = (20 - 16.66667)² / 16.66667 + (20 - 16.66667)² / 16.66667 + (10 - 16.66667)² / 16.66667

= 4.00

Degree of freedom = number of majors - 1 = 3 - 1 = 2

Critical value of X² statistic at alpha level of .05 = 5.99

Since the observed X² statistic is less than the Critical value, we fail to reject null hypothesis H0 and conclude that there is no significant evidence that the number of students choosing each category of majors differs.