A random sample of n = 16 scores is selected from a normal population with a mean of μ = 50. After a treatment is administered to the individuals in the sample, the sample mean is found to be M = 54. If the population standard deviation is σ = 8, is the sample mean sufficient to conclude that the treatment has a significant effect? Use a two-tailed test with α = .05. (Hint: Recall that the critical value for a two-tailed test with α = .05 is z < –1.96 or z > 1.96.) With σ = 8, z =    . You  ___   conclude that the treatment has a significant effect. If the population standard deviation is σ = 12, is the sample mean sufficient to conclude that the treatment has a significant effect? Use a two-tailed test with α = .05. With σ = 12, z =    . You  ____  conclude that the treatment has a significant effect.   Comparing your answers for the two cases you just tested, explain how the magnitude of the standard deviation influences the outcome of a hypothesis test. A larger standard deviation ____the likelihood of rejecting the null hypothesis.

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 + ...