Question

In Exercises 5–16, test the given claim.

Blanking Out on Tests Many students have had the unpleasant experience of panicking on atest because the first question was exceptionally difficult. The arrangement of test items was studiedfor its effect on anxiety. The following scores are measures of “debilitating test anxiety,” whichmost of us call panic or blanking out (based on data from “Item Arrangement, Cognitive EntryCharacteristics, Sex and Test Anxiety as Predictors of Achievement in Examination Performance,”by Klimko, Journal of Experimental Education, Vol. 52, No. 4.) Using a 0.05 significance level,test the claim that the two populations of scores have different amounts of variation.

Questions Arranged from Easy to Difficult
24.64	39.29	16.32	32.83	28.02
33.31	20.60	21.13	26.69	28.9
26.43	24.23	7.10	32.86	21.06
28.89	28.71	31.73	30.02	21.96
25.49	38.81	27.85	30.29	30.72

Questions Arranged from Difficult to Easy
33.62	34.02	26.63	30.26
35.91	26.68	29.49	35.32
27.24	32.34	29.34	33.53
27.62	42.91	30.20	32.54

Short answer

Thus, there is not enough evidence to support the claim that the two populations of scores have different amounts of variation.

Step-by-step solution

Given information

Two samples are considered.

One sample represents anxiety scores due to the arrangement of questions from easy to difficultin the test paper with a sample size equal to 25,and the other representsanxiety scores due to the arrangement of questions from difficult to easy in the test paper with a sample size equal to 16.

It is claimed that the variationin the scores corresponding to the arrangement of questions from easy to difficult is different from the variation in the scores corresponding to the arrangement of questions from difficult to easy.

Hypotheses

Let\({\sigma _1}\)and\({\sigma _2}\)be the population standard deviationsof the scores corresponding to the arrangement of questions from easy to difficult and the arrangement from difficult to easy, respectively.

Null Hypothesis: The population standard deviation of the scores corresponding to the arrangement of questions from easy to difficult is equal to thepopulation standard deviation of the scores corresponding to the arrangement of questions from difficult to easy.

Symbolically,

\({H_0}:{\sigma _1} = {\sigma _2}\)

Alternate Hypothesis: The population standard deviation of the scores corresponding to the arrangement of questions from easy to difficult is not equal to thepopulation standard deviation of the scores corresponding to the arrangement of questions from difficult to easy.

Symbolically,

\({H_1}:{\sigma _1} \ne {\sigma _2}\)

Sample mean, sample size, and sample variances

The sample mean score corresponding to the arrangement of questions from easy to difficultis equal to:

\(\begin{array}{c}{{\bar x}_1} = \frac{{\sum\limits_{i = 1}^{{n_1}} {{x_i}} }}{{{n_1}}}\\ = \frac{{24.64 + 39.29 + .... + 30.72}}{{25}}\\ = 27.12\end{array}\)

The varianceofthe scorescorresponding to the arrangement of questions from easy to difficult is equal to:

\(\begin{array}{c}{s_{{\rm{easy}}\;{\rm{to}}\,{\rm{difficult}}}} = \frac{{\sum\limits_{i = 1}^{{n_1}} {{{({x_i} - {{\bar x}_1})}^2}} }}{{{n_1} - 1}}\\ = \frac{{{{\left( {24.64 - 27.12} \right)}^2} + {{\left( {39.29 - 27.12} \right)}^2} + .... + {{\left( {30.72 - 27.12} \right)}^2}}}{{25 - 1}}\\ = 47.02\end{array}\)

Therefore, the variance ofthe scorescorresponding to the arrangement of questions from easy to difficult is equal to 47.02.

The mean score corresponding to the arrangement of questions fromdifficult to easy is equal to:

\(\begin{array}{c}{{\bar x}_2} = \frac{{\sum\limits_{i = 1}^{{n_2}} {{x_i}} }}{{{n_2}}}\\ = \frac{{33.62 + 34.02 + .... + 32.54}}{{16}}\\ = 31.73\end{array}\)

The variance of the scorescorresponding to the arrangement of questions fromdifficult to easy is equal to:

\(\begin{array}{c}{s_{{\rm{difficult}}\;{\rm{to}}\;{\rm{easy}}}} = \frac{{\sum\limits_{i = 1}^{{n_2}} {{{({x_i} - {{\bar x}_2})}^2}} }}{{{n_2} - 1}}\\ = \frac{{{{\left( {33.62 - 31.73} \right)}^2} + {{\left( {34.02 - 31.73} \right)}^2} + .... + {{\left( {32.54 - 31.73} \right)}^2}}}{{16 - 1}}\\ = 18.15\end{array}\)

Therefore, the variance of the scores corresponding to the arrangement of questions fromdifficult to easy is equal to 18.15.

Compute the test statistic

Since two independent samples involve a claim about the population standard deviation, apply an F-test.

Consider the larger sample variance to be\(s_1^2\)and the corresponding sample size to be\({n_1}\).

Here,\(s_1^2\)is the sample variance corresponding tothe arrangement of questions from easy to difficult is equal to 47.02.

\(s_2^2\)is the sample variance corresponding to thearrangement of questions fromdifficult to easy is equal to 18.15.

Substitute the respective values to calculate the F statistic:

\(\begin{array}{c}F = \frac{{s_1^2}}{{s_2^2}}\\ = \frac{{47.02}}{{18.15}}\\ = 2.591\end{array}\)

State the critical value and the p-value

The value of the numerator degrees of freedom is equal to:

\(\begin{array}{c}{n_1} - 1 = 25 - 1\\ = 24\end{array}\)

The value of the denominator degrees of freedom is equal to:

\(\begin{array}{c}{n_2} - 1 = 16 - 1\\ = 15\end{array}\)

For the F test, the critical value corresponding to the right-tail is considered.

The critical value can be obtained using the F-distribution table with numerator degrees of freedom equal to25and denominator degrees of freedom equal to 15 for a right-tailed test.

The level of significance is equal to:

\(\begin{array}{c}\frac{\alpha }{2} = \frac{{0.05}}{2}\\ = 0.025\end{array}\)

Thus, the critical value is equal to 2.7006

The two-tailed p-value for F equal to 2.593 is equal to 0.0596.

Conclusion

Since the test statistic value is less than the critical value and the p-value is greater than 0.05, the null hypothesis is failed to reject.

Thus, there is not enough evidence to supportthe claimthat the two populations of scores have different amounts of variation.