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An experiment was designed to test whether students’ learning is affected by background sound. Twenty-four students were randomly separated into three groups of eight. All students study a passage of text for 30 minutes.

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Group 1 study with music playing at a constant volume in the background. Group 2 study with recorded conversations playing in the background and group 3 study with no sound at all. After studying, all students take a multiple-choice test on the material.

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Their scores are given below:

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Group 1Group 2Group 3
{r1[0]}{r2[0]}{r3[0]}
{r1[1]}{r2[1]}{r3[1]}
{r1[2]}{r2[2]}{r3[2]}
{r1[3]}{r2[3]}{r3[3]}
{r1[4]}{r2[4]}{r3[4]}
{r1[5]}{r2[5]}{r3[5]}
{r1[6]}{r2[6]}{r3[6]}
{r1[7]}{r2[7]}{r3[7]}
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Test the hypothesis that the students' average marks are unaffected by the background sounds.

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Sum of squaresdegrees of freedommean squareF
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Between

\n		[[0]]2[[2]][[4]]
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Within

\n		[[1]]21[[3]]
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Correct to 2 decimal places, enter the lower table statistic: F = [[5]]

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Correct to 2 decimal places, enter the upper table statistic: F = [[6]]

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The test statistic falls in the upper critical region. Reject the Null Hypothesis at the 5% level and conclude that student's average marks are affected by the background sounds.

", "

Accept the Null Hypothesis at the 5% significance level and conclude that student's average marks are unaffected by the background sounds.

", "

The test statistic falls in the lower critical region. Reject the Null Hypothesis at the 5% level and conclude that student's average marks are affected by the background sounds.

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Having compared your test statistic with the upper and lower table values for a two-tailed F-test having 2, 21 degrees of freedom, select one of the following conclusions that best describes your conclusion.

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In this example we are comparing the means of three distict samples, we therefore must use the analysis of variance test (ANOVA)

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\\(H_0:\\) Students' average marks are unaffected by different background sounds.

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\\(H_1:\\) Students' average marks are affected by different background sounds.

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Group 1Group 2Group 3
\\({n_i}\\)888
\\(\\sum{x_i}\\){sum_r1}{sum_r2}{sum_r3}
\\(\\sum{x_i^2}\\){r1sqd}{r2sqd}{r3sqd}
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We need to evaluate the sum of squares between the samples \\(S_b\\), and the sum of squares within the samples \\(S_w\\).

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\\(S_b=\\sum\\left(\\frac{(\\sum{x_i})^2}{n_i}\\right)-\\frac{(\\sum{x_{i,j}})^2}{n_t}\\) where \\({n_i}\\) is the number of data values in sample \\(i\\) and \\({n_t}\\) equals the total number of data values.

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\\(S_b=\\frac{(\\var{sum_r1})^2}{8}+\\frac{(\\var{sum_r2})^2}{8}+\\frac{(\\var{sum_r3})^2}{8}-\\frac{(\\var{total_sum_r})^2}{24}\\)

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\\(S_b=\\var{ss_betw}\\)

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\\(S_w=\\sum\\left(\\sum{x_i^2}-\\frac{(\\sum{x_i})^2}{n_i}\\right)\\)

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\\(S_w=\\left(\\var{r1sqd}-\\frac{(\\var{sum_r1})^2}{8}\\right)+\\left(\\var{r2sqd}-\\frac{(\\var{sum_r2})^2}{8}\\right)+\\left(\\var{r3sqd}-\\frac{(\\var{sum_r3})^2}{8}\\right)\\)

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\\(S_w=\\var{ss_within}\\)

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Sum of Squaresdeg. of freedomMean squareF
Between\\(\\var{ss_betw}\\)2\\(\\var{msb}\\)\\(\\var{test_statistic}\\)
                  Within\\(\\var{ss_within}\\)21\\(\\var{msw}\\)
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The F-table values will have \\(F_{2,21}=2, 21\\) degrees of freedom.

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The lower cut-off point is found using \\(\\frac{1}{F_{21,2}}\\)

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Significance0.05
Lower limit0.025
Upper limit4.42
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Compare the test statistic with the F-table values and choose your conclusion.

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