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Description: A recent study of 100 social workers in a southern state found that their mean salary was $23,000 with a standard deviation of $2100. A similar study of 95 social workers in a western state found that their mean salary was $34,200 with a standard dev

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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A recent study of 100 social workers in a southern state found that their mean salary was $23,000 with a standard deviation of $2100. A similar study of 95 social workers in a western state found that their mean salary was $34,200 with a standard deviation of$3200. Find the test statistic and critical value when the claim that the mean salaries of social workers in the western state is greater than that of social workers in the southern state. Use α = 0.05.

Sample1: Social workers in a southern state, Sample2: Social workers in awestern state.

a) Find the test statistic

b) Find critical value

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

a). Sample1-

sample size= $n_{1} = 100$

sample mean=$23000= $x_{1}$

population mean= $μ_{1}$

$s_{1} = $ 2100$

Sample2-

sample size= $n_{2} = 95$

sample mean= $x_{2}$ =$34200

population mean= $μ_{2}$

$s_{2} = $ 3200$

hypotheses to be tested here will be,

$H_{0} : μ_{1} = μ_{2} v s H_{2} : μ_{1} < μ_{2}$

So, the test statistic will be given by,

$t = \frac{x_{1} - x_{2} - (μ_{1} - μ_{2})}{s \sqrt{\frac{1}{n_{1}} + \frac{1}{n_{2}}}}$

where

$s^{2} = \frac{(n_{1} - 1) s_{1}^{2} + (n_{2} - 1) s_{2}^{2}}{n_{1} + n_{2} - 2} = 7249481.865$

Therefore, s=2692.486

Value of t under null hypothesis is -29.029

So, $|t| = 29.029$

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