Researchers conducted a study to determine whether magnets are effective in treating back pain. The results are shown for the treatment​ (with magnets) group and the sham​ (or placebo) group. Treatment with sample size of 42 and mean of 0.69 and standard deviation of 0.25 Sham with sample size of 44 and mean of 0.75 and standard deviation of 0.44 The results are a measure of reduction in back pain. Assume that the two samples are independent simple random samples selected from normally distributed​ populations, and do not assume that the population standard deviations are equal. Test the claim that the treatment mean measure of reduction in back pain is greater than the sham mean measure of reduction in back pain. Use the Critical Value Method. Show all the work on paper and enter the critical value with 3-decimal places.  show formulas and draw all curves.

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...