A random sample of n = 12 individuals is selected from a population with µ = 70, and a treatment is administered to each individual in the sample. After treatment, the sample mean is found to be M = 74.5 with SS = 297. Degrees of Freedom = 21 1. How much difference is there between the mean for the treated sample and the mean for the original population? (Note: In a hypothesis test, this value forms the numerator of the t statistic.) 2. If there is no treatment effect, how much difference is expected just by chance between the sample mean and its population mean? That is, find the standard error for M. (Note: In a hypothesis test, this value is the denominator of the t statistic.) 3. Based on the sample data, does the treatment have a significant effect? Use a two-tailed test with α = .05. (Show three decimal places.) t-critical = ± t = The results indicate: a. Failure to reject the null hypothesis; there is a significant treatment effect. b. Rejection of the null hypothesis; there is a significant treatment effect. c. Failure to reject the null hypothesis; there is not a significant treatment effect. d. Rejection of the null hypothesis; there is not a significant treatment effect.

Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018

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Chapter10: Statistics

Section10.1: Measures Of Center

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8. Gravetter/Wallnau/Forzano, Essentials - Chapter 9 - End-of-chapter question 9

A random sample of n = 12 individuals is selected from a population with µ = 70, and a treatment is administered to each individual in the sample. After treatment, the sample mean is found to be M = 74.5 with SS = 297.

Degrees of Freedom = 21

1. How much difference is there between the mean for the treated sample and the mean for the original population? (Note: In a hypothesis test, this value forms the numerator of the t statistic.)

2. If there is no treatment effect, how much difference is expected just by chance between the sample mean and its population mean? That is, find the standard error for M. (Note: In a hypothesis test, this value is the denominator of the t statistic.)

3. Based on the sample data, does the treatment have a significant effect? Use a two-tailed test with α = .05. (Show three decimal places.)

t-critical	=	±
t	=

The results indicate:

a. Failure to reject the null hypothesis; there is a significant treatment effect.

b. Rejection of the null hypothesis; there is a significant treatment effect.

c. Failure to reject the null hypothesis; there is not a significant treatment effect.

d. Rejection of the null hypothesis; there is not a significant treatment effect.

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 + ...

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Denote μ as the true population mean, M as the sample mean, SS as the sum of squares from the mean for the sample, and s as the sample standard deviation.

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