Question

A random sample of 100 high school students was surveyed regarding their favorite subject. The following counts were obtained: Favorite Subject Number of Students English Math Science 30 Art/Music The researcher conducted a test to determine whether the proportion of students was equal for all four subjects. What is the value of the test statistic? O b. 25 OOOO d. -4 How many degrees of freedom does the chi-square test statistic for a goodness of fit have when there are 10 categories? a. 74 OOOO d. 62

Answer 1

a)
`\chi^2 = ((25-25)^2)/(25)+((30-25)^2)/(25)+((30-25)^2)/(25)+((15-25)^2)/(25)=6`

b)
`df=Categories-1=10-1=9`

Explanation:

We assume the following info:

Favorite Subject Number of students

English 25

Math 30

Science 30

Art/Music 15

Total 100

Previous concepts

A chi-square goodness of fit test "determines if a sample data matches a population".

A chi-square test for independence "compares two variables in a contingency table to see if they are related. In a more general sense, it tests to see whether distributions of categorical variables differ from each another".

Part a

The system of hypothesis on this case are:

H0: There is no difference with the distribution proposed

H1: There is a difference with the distribution proposed

The level os significance assumed for this case is
`\alpha=0.05`

The statistic to check the hypothesis is given by:

`\chi^2 =\sum_(i=1)^n ((O_i -E_i)^2)/(E_i)`

The table given represent the observed values, we just need to calculate the expected values are 25 for each category.

And the calculations are given by:

`E_(English) =25`

`E_(Math) =25`

`E_(Science) =25`

`E_(Music) =25`

And now we can calculate the statistic:

`\chi^2 = ((25-25)^2)/(25)+((30-25)^2)/(25)+((30-25)^2)/(25)+((15-25)^2)/(25)=6`

Now we can calculate the degrees of freedom for the statistic given by:

`df=Categories-1=4-1=3`

And we can calculate the p value given by:

`p_v = P(\chi^2_(3) >6)=0.112`

And we can find the p value using the following excel code:

"=1-CHISQ.DIST(6,3,TRUE)"

Part b

For this case we have this formula:

`df=Categories-1=10-1=9`