Question

A high school runs a survey asking students if they participate in sports. The results are found below. Run an independence test for the data at α=0.01. Freshmen Sophomores Juniors Seniors Yes 75 88 55 42 No 30 28 38 40 Can it be concluded that participation in sports is dependent on grade level?

Answer 1

Answer:

It can be concluded that participation in sports is dependent on grade level.

Step-by-step explanation:

In this case a Chi-square independence test for the data is to be performed at α = 0.01.

The hypothesis can be defined as follows:

H₀: The participation in sports is independent of grade level.

Hₐ: The participation in sports is dependent of grade level.

The data provided is:

            Freshmen      Sophomores       Juniors         Seniors

Yes            75                       88                   55                42

No             30                      28                   38                 40

The formula to compute the expected frequencies is:

$E_{i}=\frac{i^{th}\ \text{Row Total}\ \times\ i^{th}\ \text{Column Total}}{N}$

The Chi-square statistic is:

$\chi^{2}=\sum {\frac{(O_{i}-E_{i})^{2}}{E_{i}}}$

Consider the Excel file attached below.

The value of Chi-square statistic is 16.244.

The degrees of freedom of the test are:

$\text{df}=(r-1)(c-1)$

    $=(4-1)(2-1)\\=3\times 1\\=3$

Compute the p-value of the test as follows:

$p-value=P(\chi^{2}_{df}<\chi^{2})$

                $=P(\chi^{2}_{3}<16.244)\\=0.001$

*Use a Chi-square table.

p-value = 0.001 < α = 0.01

The null hypothesis will be rejected at 1% level of significance.

Thus, it can be concluded that participation in sports is dependent on grade level.