Asked by Vanessa Ramirez on Jun 03, 2024
Verified
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 from the university 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 hypothesis is to be tested at the 5% level of significance.The critical value from the table equals
A) 7.378.
B) 9.348.
C) 5.991.
D) 7.815.
Critical Value
A point on the scale of the test statistic beyond which we reject the null hypothesis, significant in hypothesis testing.
Level of Significance
The probability of rejecting the null hypothesis in a statistical test when it is actually true; a measure of the risk of making a Type I error.
Proportions
A statistical measure that reflects the size of a part compared to the whole, typically expressed as a fraction or a percentage.
- Evaluate the calculation and implications of test statistics for hypothesis verification.
- Carry out hypothesis examinations for proportions and decipher the conclusions.
Verified Answer
ZK
Zybrea KnightJun 08, 2024
Final Answer :
C
Explanation :
To test if the proportions have changed, we will use a chi-square goodness-of-fit test.
The null hypothesis is that the proportions of students in each college have not changed:
H0: p1 = 0.35, p2 = 0.35, p3 = 0.30
The alternative hypothesis is that the proportions have changed:
Ha: at least one proportion has changed
We will use a significance level of 0.05.
The chi-square test statistic is calculated as:
χ2 = Σ [(O - E)2 / E]
where O is the observed frequency and E is the expected frequency under the null hypothesis.
Expected frequencies can be calculated as:
E1 = 0.35 * 300 = 105
E2 = 0.35 * 300 = 105
E3 = 0.30 * 300 = 90
χ2 = [(90 - 105)2 / 105] + [(120 - 105)2 / 105] + [(90 - 90)2 / 90]
= 2.143 + 2.143 + 0
= 4.286
The degrees of freedom for this test is (3 - 1) = 2.
Looking at the chi-square distribution table with 2 degrees of freedom and a significance level of 0.05, the critical value is 5.991.
Since our calculated chi-square test statistic (4.286) is less than the critical value (5.991), we fail to reject the null hypothesis. We do not have enough evidence to conclude that the proportions of students in each college have changed.
The null hypothesis is that the proportions of students in each college have not changed:
H0: p1 = 0.35, p2 = 0.35, p3 = 0.30
The alternative hypothesis is that the proportions have changed:
Ha: at least one proportion has changed
We will use a significance level of 0.05.
The chi-square test statistic is calculated as:
χ2 = Σ [(O - E)2 / E]
where O is the observed frequency and E is the expected frequency under the null hypothesis.
Expected frequencies can be calculated as:
E1 = 0.35 * 300 = 105
E2 = 0.35 * 300 = 105
E3 = 0.30 * 300 = 90
χ2 = [(90 - 105)2 / 105] + [(120 - 105)2 / 105] + [(90 - 90)2 / 90]
= 2.143 + 2.143 + 0
= 4.286
The degrees of freedom for this test is (3 - 1) = 2.
Looking at the chi-square distribution table with 2 degrees of freedom and a significance level of 0.05, the critical value is 5.991.
Since our calculated chi-square test statistic (4.286) is less than the critical value (5.991), we fail to reject the null hypothesis. We do not have enough evidence to conclude that the proportions of students in each college have changed.
Learning Objectives
- Evaluate the calculation and implications of test statistics for hypothesis verification.
- Carry out hypothesis examinations for proportions and decipher the conclusions.