Question

at a school pep rally, a group of sophomore students organized a free raffle for prizes. they claim that they put the names of all of the students in the school in the basket and that they randomly drew 36 names out of this basket. of the prize winners, 6 were freshmen, 14 were sophomores, 9 were juniors, and 7 were seniors. the results do not seem that random to you. you think it is a little fishy that sophomores organized the raffle and also won the most prizes. your school is composed of 30% freshmen, 25% sophomores, 25% juniors, and 20% seniors. a. what are the expected frequencies of winners from each class? b. conduct a significance test to determine whether the winners of the prizes were distributed throughout the classes as would be expected based on the percentage of students in each group. report your chi square and p values.

Answer 1

The winners of the prizes are distributed throughout the classes as per the percentage of students in the respective group.

The total number of surveyed students is 36.

We calculate the expected number of students for each class is:

Class

Expected Frequency

Freshman

30% of 36 = 10.8

Sophomore

25% of 36 = 9

Junior

25% of 36 = 9

Senior

20% of 36 = 7.2

(b)

We set up:

H0: The number of observed winners is indifferent to the number of expected winners.

H1: The number of observed winners is not indifferent to the number of expected winners.

The Test Statistic Calculation Table:

Class Observed Frequency (fi) Expected Frequency (ei) (fi -ei )² /ei

Freshman 6                                   10.8                                 2.1333

Sophomore 14                                    9                                         2.7778

Junior         9                                    9                                           0

Senior         7                                    7.2                                0.0056

Total =        36                                    36                                    4.9167

The calculated

$x^{2}$ = 4.9167

The p-value for 4 - 1=3 degree of freedom for the above test score is 0.177999.

(c)

Since p-value (=0.177999) >  α0..05 (standard level of significance),

H0 is failed to be rejected.

The number of observed winners is concluded to be indifferent to the number of expected winners, i.e., the winners of the prizes are distributed throughout the classes as per the percentage of students in the respective group.

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