Magnet Treatment of Pain Researchers conducted a study to determine whether magnets are effective in treating back pain, with results given below (based on data from “Bipolar Permanent Magnets for the Treatment of Chronic Lower Back Pain: A Pilot Study,” by Collacott, Zimmerman, White, and Rindone, Journal of the American Medical Association, Vol. 283, No. 10). The values represent measurements of pain using the visual analog scale. Use a 0.05 significance level to test the claim that those given a sham treatment (similar to a placebo) have pain reductions that vary more than the pain reductions for those treated with magnets.

Reduction in Pain Level After Sham Treatment: n = 20, \(\bar x\) = 0.44, s = 1.4

Reduction in Pain Level After Magnet Treatment: n = 20, \(\bar x\) = 0.49, s = 0.96

In a sample of 20 people given the sham treatment, the mean reduction in pain level is equal to 0.44,and the standard deviation of the reduction in pain level is equal to 1.4. In another sample of 20 people given the magnet treatment, the mean reduction in pain levelequals0.49,and the standard deviation of the reduction in pain levelequals0.96.

It is claimed that the variation in the pain reduction for those given a sham treatment is more than the variation in the pain reduction for those given the magnet treatment.

Let\({\sigma _1}\)and\({\sigma _2}\)be the population standard deviationsof the reduction in pain level corresponding to the sham treatment and the magnet treatment,respectively.

Nullhypothesis:The populationstandard deviationof the reduction in pain level for those given the sham treatment equalsthe population standard deviation of the reduction in pain level corresponding to the magnet treatment.

Symbolically,

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

Alternativehypothesis:The populationstandard deviationof the reduction in pain level for those who were given the sham treatment is greater than the population standard deviation of the reduction in pain level for those who were given the magnet treatment.

Symbolically,

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

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\({n_1}\).

The following values are obtained:

\({\left( {1.4} \right)^2} = 1.96\)

\({\left( {0.96} \right)^2} = 0.9216\)

Here,\(s_1^2\)is the sample variance corresponding to the sham treatment and has a value equal to 1.96.

\(s_2^2\)is the sample variance corresponding to the magnet treatment and has a value equal to 0.9216.

Substitute the respective values to calculate the F statistic:

\(\begin{array}{c}F = \frac{{s_1^2}}{{s_2^2}}\\ = \frac{{{{\left( {1.4} \right)}^2}}}{{{{\left( {0.96} \right)}^2}}}\\ = 2.127\end{array}\)

Thus, the value of F is equal to 2.127.

The value of the numerator degrees of freedomequals the following:

\(\begin{array}{c}{n_1} - 1 = 20 - 1\\ = 19\end{array}\)

The value of the denominator degrees of freedomequals the following:

\(\begin{array}{c}{n_2} - 1 = 20 - 1\\ = 19\end{array}\)

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 to 19 and denominator degrees of freedom equal to 19 for a right-tailed test.

The level of significance is equal to 0.05.

Thus, the critical value is equal to 2.1683.

The right-tailed p-value for F equal to 2.127 is equal to 0.0543.

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 claimthatthe variation in the pain reduction for those given a sham treatment is more than the variation in the pain reduction for those given the magnet treatment.