In: Statistics and Probability
Professor Stone complains that student teacher ratings depend on the grade the student receives. In other words, according to Professor Stone, a teacher who gives good grades gets good ratings, and a teacher who gives bad grades gets bad ratings. To test this claim, the Student Assembly took a random sample of 300 teacher ratings on which the student's grade for the course also was indicated. The results are given in the following table. Test the hypothesis that teacher ratings and student grades are independent at the 0.01 level of significance.
RatingABCF (or withdrawal)Row Total
Excellent151715552
Average25307012137
Poor23264319111
Column Total637312836300
(i) Give the value of the level of significance.
State the null and alternate hypotheses.
H0: The distributions for the different ratings are the
same.
H1: The distributions for the different ratings are different.H0:
Ratings of excellent, average, and poor are independent.
H1: Ratings of excellent, average, and poor are not
independent. H0: Student grade and teacher
rating are independent.
H1: Student grade and teacher rating are not independent.H0: Tests
A, B, C, F (or withdrawal) are independent.
H1: Tests A, B, C, F (or withdrawal) are not independent.
(ii) Find the sample test statistic. (Round your answer to two
decimal places.)
(iii) Find or estimate the P-value of the sample test
statistic.
P-value > 0.1000.050 < P-value < 0.100 0.025 < P-value < 0.0500.010 < P-value < 0.0250.005 < P-value < 0.010P-value < 0.005
(iv) Conclude the test.
Since the P-value ≥ α, we reject the null hypothesis.Since the P-value ≥ α, we do not reject the null hypothesis. Since the P-value < α, we do not reject the null hypothesis.Since the P-value < α, we reject the null hypothesis.
(v) Interpret the conclusion in the context of the application.
At the 1% level of significance, there is insufficient evidence to claim that student grade and teacher rating are not independent.At the 1% level of significance, there is sufficient evidence to claim that student grade and teacher rating are not independent.
using excel>addin<phstat>multiple sample test
we have
Chi-Square Test | ||||||||||
Observed Frequencies | ||||||||||
Column variable | Calculations | |||||||||
Row variable | A | B | C | F | Total | fo-fe | ||||
Excellent | 15 | 17 | 15 | 5 | 52 | 4.08 | 4.346667 | -7.18667 | -1.24 | |
Avergae | 25 | 30 | 70 | 12 | 137 | -3.77 | -3.33667 | 11.54667 | -4.44 | |
Poor | 23 | 26 | 43 | 19 | 111 | -0.31 | -1.01 | -4.36 | 5.68 | |
Total | 63 | 73 | 128 | 36 | 300 | |||||
Expected Frequencies | ||||||||||
Column variable | ||||||||||
Row variable | A | B | C | F | Total | (fo-fe)^2/fe | ||||
Excellent | 10.92 | 12.65333 | 22.18667 | 6.24 | 52 | 1.524396 | 1.493165 | 2.327893 | 0.24641 | |
Avergae | 28.77 | 33.33667 | 58.45333 | 16.44 | 137 | 0.494018 | 0.333967 | 2.280888 | 1.199124 | |
Poor | 23.31 | 27.01 | 47.36 | 13.32 | 111 | 0.004123 | 0.037767 | 0.401385 | 2.422102 | |
Total | 63 | 73 | 128 | 36 | 300 | |||||
Data | ||||||||||
Level of Significance | 0.01 | |||||||||
Number of Rows | 3 | |||||||||
Number of Columns | 4 | |||||||||
Degrees of Freedom | 6 | |||||||||
Results | ||||||||||
Critical Value | 16.81189 | |||||||||
Chi-Square Test Statistic | 12.76524 | |||||||||
p-Value | 0.046919 | |||||||||
Do not reject the null hypothesis |
(i) the value of the level of significance is 0.01
the null and alternate hypotheses.
H0: Student grade and teacher rating are independent.
H1: Student grade and teacher rating are not independent.
(ii) the sample test statistic=12.77
(iii) Find or estimate the P-value of the sample test
statistic.
0.025 < P-value < 0.050
(iv) Conclude the test.
Since the P-value ≥ α, we do not reject the null hypothesis.
(v) Interpret the conclusion in the context of the application.
At the 1% level of significance, there is insufficient evidence to claim that student grade and teacher rating are not independent