Question

9.
A school social worker wants to determine if the grade distribution of home-schooled children is different in her
district than nationally. A national statistical center provided her with the data in the first table below, which
represent the relative frequency of home-schooled children by grade level. She obtains a sample of
home-schooled children within her district that yields the data in the second table below.
Grade Relative Frequency Grade Frequency
K 0.045 K 4
1-3 0.256 1-3 9
4-5 0.122 4-5 2
6-8 0.274 6-8 6
9-12 0.303 9-12 4

(a) Because of the low cell counts, combine cells into three categories K-3, 4-8, and 9-12.
Grade Relative Frequency Observed Frequency Expected Frequency
K-3 ? ? ?
4-8 ? ? ?
9-12 ? ? ?
(Type integers or decimals rounded to three decimal places as needed.)

(b) Is the grade distribution of home-schooled children different in her district from the national grade distribution
at the α = 0.01 level of significance?

What are the hypotheses?
(1) H₀ or H₁ : The grade distribution of home-schooled children in her district is the national
grade distribution of home-schooled children.

(2) H₀ or H₁ : The grade distribution of home-schooled children in her district is the national
grade distribution of home-schooled children.

Use technology to compute the P-value for this test at α = 0.01 level of significance.
P-value = (Round to three decimal places as needed.)

Find the conclusion to the hypothesis test.
(3) reject / do not reject the null hypothesis.

There (4) is / is not sufficient evidence at the level of
significance to conclude that the grade distribution of home-schooled children in her district is (5) the same as / different from
the national grade distribution of home-schooled children.

Answer 1

9.

(a)

(b)

We have to perform Chi-square test for goodness of fit.

We have to test for null hypothesis

The grade distribution of home-schooled children in her district is same as the national grade distribution of home-schooled children.

against the alternative hypothesis

The grade distribution of home-schooled children in her district is not same as the national grade distribution of home-schooled children.

Our Chi-square test statistic is given by

Here,

Number of classes

Under null hypothesis, expected frequencies and necessary calculations are as follows.

Degrees of freedom

[Using R-code '1-pchisq(6.035,2)']

Level of significance

We reject our null hypothesis if

Here, we observe that

So, we cannot reject our null hypothesis.

Hence, based on the given data we can conclude that there is not sufficient evidence at the 0.01 level of significance to conclude that the grade distribution of home-schooled children in her district is different from the national grade distribution of home-schooled children.