How do you find the p-value???

Thanks!

In the past, 35% of the students at ABC University were in the Business College, 35% of the students were in the Liberal Arts College, and 30% of the students were in the Education College. To see whether or not the proportions have changed, a sample of 300 students was taken. Ninety of the sample students are in the Business College, 120 are in the Liberal Arts College, and 90 are in the Education College.

The p-value is

a. greater than 0.1

b. between 0.05 and 0.1

c. between 0.025 and 0.05

d. between 0.01 and .025

To calculate the p-value in this scenario, we can use a chi-square test for independence. This test helps determine if the distribution of categorical variables is independent or dependent.

Here are the steps to calculate the p-value:

Step 1: Set up hypotheses
The null hypothesis (H0) assumes that the proportions of students in each college have not changed. The alternative hypothesis (Ha) assumes that the proportions have changed.

H0: The proportions have not changed.
Ha: The proportions have changed.

Step 2: Calculate the expected frequencies
To compare observed frequencies with expected frequencies, we need to calculate the expected frequencies assuming the null hypothesis is true. To do this, multiply the overall proportions by the total sample size.

Expected frequency for Business College = (0.35) * 300 = 105
Expected frequency for Liberal Arts College = (0.35) * 300 = 105
Expected frequency for Education College = (0.3) * 300 = 90

Step 3: Compute the chi-square test statistic
To calculate the chi-square test statistic, use the formula:

χ² = Σ [(Observed frequency - Expected frequency)² / Expected frequency]

Calculate this for each category and sum the values:

χ² = [(90-105)²/105] + [(120-105)²/105] + [(90-90)²/90]

Step 4: Determine the degrees of freedom (df)
The degrees of freedom for a chi-square test of independence are calculated using the formula:

df = (number of rows - 1) * (number of columns - 1)

In this case, there are 3 categories (colleges) and 3-1 = 2 degrees of freedom.

Step 5: Find the p-value
Using the chi-square distribution table or a calculator, find the p-value corresponding to the calculated chi-square test statistic and degrees of freedom.

The p-value can then be compared to a significance level (usually 0.05) to make a decision. In this case, we are given multiple choices for the p-value range.

From the calculations, the test statistic will have a p-value greater than 0.1. Therefore, the correct answer is (a) greater than 0.1.