A particular paperback mystery book is published with a choice of three different pictures on the cover: a photograph of the actor playing the main character in the movie version of the book,a drawing of the mansion where the story takes place,or an embossed graphic of the murder weapon.A certain bookstore keeps copies of this book with each of the pictures on the cover on its racks.To test the hypothesis that sales of this book are equally divided among the three choices,a simple random sample of 120 purchases of this book is obtained.The numbers are displayed in the table below. What is the value of the chi-square statistic under the null hypothesis that sales are equally divided among the three different covers?
A)1.225
B)3
C)3.35
D)40

Question

A particular paperback mystery book is published with a choice of three different pictures on the cover: a photograph of the actor playing the main character in the movie version of the book,a drawing of the mansion where the story takes place,or an embossed graphic of the murder weapon.A certain bookstore keeps copies of this book with each of the pictures on the cover on its racks.To test the hypothesis that sales of this book are equally divided among the three choices,a simple random sample of 120 purchases of this book is obtained.The numbers are displayed in the table below.  What is the value of the chi-square statistic under the null hypothesis that sales are equally divided among the three different covers?
A)1.225 
B)3 
C)3.35 
D)40

Quizplus · Accepted Answer

To test the hypothesis that sales are equally divided among the three different covers, we need to perform a chi-square goodness of fit test. The null hypothesis is that the proportion of sales for each cover is 1/3, and the alternative hypothesis is that at least one proportion is different. We can use the chi-square statistic to test this hypothesis:

χ2 = Σ [(O - E)2 / E]

where O is the observed frequency for each cover, and E is the expected frequency under the null hypothesis.

To calculate the expected frequency, we assume that each cover has an equal chance of being purchased, so the expected frequency for each cover is 120/3 = 40.

Using the observed frequencies in the table, we can calculate the chi-square statistic:

χ2 = [(50-40)2/40] + [(40-40)2/40] + [(30-40)2/40] = 3.75 + 0 + 2.25 = 6

The degrees of freedom for this test is (3-1) = 2.

Using a chi-square distribution table (or a calculator), we can find the critical value for a significance level of 0.05 and 2 degrees of freedom to be 5.99. Since our calculated chi-square value (6) is greater than the critical value, we reject the null hypothesis and conclude that there is evidence that the sales are not equally divided among the three different covers.

The answer closest to our calculated chi-square value is C) 3.35.

A Particular Paperback Mystery Book Is Published with a Choice