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Name: A researcher is evaluating the influence of a treatment using a sample
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Description: A researcher is evaluating the influence of a treatment using a sample selected from a normally distributed population with a mean of μ = 50 and a standard deviation of σ = 10. The researcher expects a +5-point treatment effect and plans to use a two

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

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A researcher is evaluating the influence of a treatment using a sample selected from a normally distributed population with a mean of μ = 50 and a standard deviation of σ = 10. The researcher expects a +5-point treatment effect and plans to use a two-tailed hypothesis test with α = .05.

Standard Normal Distribution

Mean = 0.0

Standard Deviation = 1.0

-2.0-1.00.01.02.0z.5000.50000.00

Compute the power of the test if the researcher uses a sample of n = 4 individuals.

The power for the test is the probability of obtaining a z-score than

, which corresponds to p =

. Thus, the power of the test is

Compute the power of the test if the researcher uses a sample of n = 25 individuals.

The power for the test is the probability of obtaining a z-score than

, which corresponds to p =

. Thus, the power of the test is

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 + ...

Expert Solution

Step 1

Given Information:

Population mean $μ = 50$

Population standard deviation $σ = 10$

The researcher expects a +5-point treatment effect.

Significance level $α = 0.05$ (two tailed test)

To compute the power of the test if the researcher uses a sample of n = 4 individuals:

Calculate the parameters of null hypothesis $H_{0}$ is true, sampling distribution:

Mean of sampling distribution: $μ_{M} = μ = 50$

Standard deviation of sampling distribution $σ_{M} = \frac{σ}{\sqrt{n}} = \frac{10}{\sqrt{4}} = 5$

Now, calculate the parameter of $H_{0}$ is false:

$μ = 50 + 5 = 55$

$σ_{M} = 5$

Calculate the critical z-score:

Using standard normal table, critical values for a two tailed test at 0.05 significance level are: -1.96 and +1.96

Raw score (M) = $μ + z σ = 50 + 1.96 (5) = 59.8$

Power = area to the right of the critical value z under $H_{0}$ is false distribution.

Location of critical z in $H_{0}$ is false distribution = $z_{m e a n o f t r e a t m e n t s} - z_{c r i t i c a l}$

$\begin{array}{rcl} z & = & \frac{M - μ}{σ_{M}} \\ = & \frac{59.8 - 55}{5} \\ = & 0.96 \end{array}$

Using standard normal table, value corresponding to 0.96 is 0.83147.

$\begin{array}{rcl} P (z > 0.96) & = & 1 - P (z \leq 0.96) \\ = & 1 - 0.83147 \\ = & 0.16853 \end{array}$

Thus, with a sample of 4 people and 5 points treatment effect, 16.853% of the time the hypothesis test will conclude that there is a significant effect.

Power of the hypothesis is 0.1685

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