Question

is achieving a basic skill level in a subject independent to the location of the school? a random sample of students categorized by the location of school and the number achieving basic skills in three subjects is shown in the table.

two-way table: location vs subject

location of school	subject	subject	subject
urban	43	42	38
suburban	63	66	65

conduct a chi-square test of independence at the 1% significance level to test whether the two variables are independent and state your conclusion.

hint: the test statistic is: \\(\chi^2 = 0.30\\)

\\(\bigcirc\\) cannot be determined.
\\(\bigcirc\\) there is not sufficient sample evidence to suggest that skill level in a subject is dependent of location.
\\(\bigcirc\\) there is sufficient sample evidence to suggest that skill level in a subject is independent of location.
\\(\bigcirc\\) there is not sufficient sample evidence to suggest that skill level in a subject is independent of location.
\\(\bigcirc\\) there is sufficient sample evidence to suggest that skill level in a subject is dependent of location.

Explanation:

Step1: Determine Degrees of Freedom

The contingency table has \( r = 2 \) rows (Urban, Suburban) and \( c = 3 \) columns (Reading, Math, Science). Degrees of freedom \( df=(r - 1)(c - 1)=(2 - 1)(3 - 1)=2 \).

Step2: Find Critical Value

For a chi - square test at \( \alpha = 0.01 \) and \( df = 2 \), from chi - square distribution table, the critical value \( \chi^2_{0.01,2}=9.210 \).

Step3: Compare Test Statistic and Critical Value

The test statistic \( \chi^2 = 0.30 \). Since \( 0.30<9.210 \), we fail to reject the null hypothesis (null hypothesis for chi - square test of independence is that the two variables are independent). Failing to reject the null hypothesis means there is not sufficient sample evidence to suggest that the variables are dependent (i.e., skill level in a subject is dependent on location).

Answer:

There is not sufficient sample evidence to suggest that skill level in a subject is dependent of location.