Question

Student performance across discussion sections: A professor who teaches a large introductory statistics class (197 students) with eight discussion sections would like to test if student performance differs by discussion section, where each discussion section has a different teaching assistant. The summary table below shows the average final exam score for each discussion section as well as the standard deviation of scores and the number of students in each section.


Sec 1 Sec 2 Sec 3 Sec 4 Sec 5 Sec 6 Sec 7 Sec 8
ni 33 19 10 29 33 10 32 31
Xi 92.94 91.11 91.8 92.45 89.3 88.3 90.12 93.35
si 4.21 5.58 3.43 5.92 9.32 7.27 6.93 4.57


The ANOVA output below can be used to test for differences between the average scores from the different discussion sections.


Df Sum Sq Mean sq F value Pr(>F)
Section 7 525.01 75 1.87
Residuals 189 7584.11 40.13


Conduct a hypothesis test to determine if these data provide convincing evidence that the average score varies across some (or all) groups. Check conditions and describe any assumptions you must make to proceed with the test.

Answer 1

Answer:

The test is not significant at 5% level of significance, hence we conclude that there's no variation among the discussion sections.

Step-by-step explanation:

Assumptions:

1. The sampling from the different discussion sections was independent and random.

2. The populations are normal with means and constant variance

$H_0:$ There's no variation among the discussion sections

$H_1:$ There's variation among the discussion sections

$\alpha =0.05$

                Df     Sum Sq       Mean sq    F value     Pr(>F)

Section     7       525.01         75             1.87            0.99986

Residuals  189    7584.11        40.13

Test Statistic = $F= \frac{75}{40.13} =1.87$

$Pr(>F) = 0.99986$

Since our p-value is greater than our level of significance (0.05), we do not reject the null hypothesis and conclude that there's no significant variation among the eight discussion sections.

Answer 2

Comparing the Critical and Statistic value of F, we can conclude that there is no variation in the average scores across the groups

The Hypothesis :

$H_{0} :$ Average score is the same across the groups

$H_{1} :$ Average score varies across the groups

From the ANOVA output given, the Fstatistic = 1.87

Decison Region :

• Reject $H_{0} :$, if Fstatistic is greater than Fcritical

Using a F distribution table or calculator ;

Df_numerator = 7 ; df_denominator = 189

Fcritical at 5% = 2.058

Hence,

Since Fstatistic < Fcritical ; we fail to reject $H_{0}$

Hence, we conclude that there is no variation in the average scores across the groups.

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