In the past,of all the students enrolled in "Basic Business Statistics" 10% earned A's 20% earned B's,30% earned C's,20% earned D's and the rest either failed or withdrew from the course.Dr Johnson is a new professor teaching "Basic Business Statistics" for the first time this semester.At the conclusion of the semester,in Dr.Johnson's class of 60 students,there were 10 A's,20 B's,20 C's,5 D's and 5 W's or F's.Assume that Dr.Johnson's class constitutes a random sample.Dr Johnson wants to know if there is sufficient evidence to conclude that the grade distribution of his class is different than the historical grade distribution. Calculate the chi-square statistic.
A)9.5
B)3.03
C)2.33
D)16.4

Question

Quizplus · Accepted Answer

First, we need to calculate the expected frequencies for each grade category based on the historical grade distribution.

Expected frequency for A's = 0.1 x 60 = 6 
Expected frequency for B's = 0.2 x 60 = 12 
Expected frequency for C's = 0.3 x 60 = 18 
Expected frequency for D's = 0.2 x 60 = 12 
Expected frequency for F's or W's = 1 - (0.1 + 0.2 + 0.3 + 0.2) = 0.2, so the expected frequency for F's or W's is 0.2 x 60 = 12

Next, we can set up a contingency table to compare the observed and expected frequencies:

A        B        C        D        F/W 
Observed   10       20     20      5        5 
Expected     6        12     18      12      12

Using the formula for the chi-square statistic, we can calculate:

χ2 = Σ [(Observed - Expected)2 / Expected]

χ2 = [(10-6)2/6] + [(20-12)2/12] + [(20-18)2/18] + [(5-12)2/12] + [(5-12)2/12]

χ2 = 16.4

Since the degrees of freedom for this test is 4 (number of categories minus 1), we can use a chi-square distribution table to find the critical value at a significance level of 0.05. The critical value is 9.488.

Since our calculated chi-square value (16.4) is greater than the critical value (9.488), we can reject the null hypothesis and conclude that there is sufficient evidence to suggest that the grade distribution in Dr. Johnson's class is different than the historical grade distribution.

In the Past,of All the Students Enrolled in "Basic Business