Answer:
The value of the test statistic is ![z = 1.78](https://tex.z-dn.net/?f=z%20%3D%201.78)
Step-by-step explanation:
Before finding the test statistic, we need to understand the central limit theorem and subtraction of normal variables.
Central Limit Theorem
The Central Limit Theorem establishes that, for a normally distributed random variable X, with mean
and standard deviation
, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
and standard deviation
.
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
Subtraction between normal variables:
When two normal variables are subtracted, the mean is the difference of the means, while the standard deviation is the square root of the sum of the variances.
Sample 1:
![\mu_1 = 110, s_1 = \frac{7.2}{\sqrt{81}} = 0.8](https://tex.z-dn.net/?f=%5Cmu_1%20%3D%20110%2C%20s_1%20%3D%20%5Cfrac%7B7.2%7D%7B%5Csqrt%7B81%7D%7D%20%3D%200.8)
Sample 2:
![\mu_2 = 108, s_2 = \frac{6.3}{\sqrt{64}} = 0.7875](https://tex.z-dn.net/?f=%5Cmu_2%20%3D%20108%2C%20s_2%20%3D%20%5Cfrac%7B6.3%7D%7B%5Csqrt%7B64%7D%7D%20%3D%200.7875)
The test statistic is:
![z = \frac{X - \mu}{s}](https://tex.z-dn.net/?f=z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7Bs%7D)
In which X is the sample mean,
is the value tested at the null hypothesis, and s is the standard error.
0 is tested at the null hypothesis:
This means that ![\mu = 0](https://tex.z-dn.net/?f=%5Cmu%20%3D%200)
Distribution of the difference:
![X = \mu_1 - \mu_2 = 110 - 108 = 2](https://tex.z-dn.net/?f=X%20%3D%20%5Cmu_1%20-%20%5Cmu_2%20%3D%20110%20-%20108%20%3D%202)
![s = \sqrt{s_1^2+s_2^2} = \sqrt{0.8^2+0.7875^2} = 1.1226](https://tex.z-dn.net/?f=s%20%3D%20%5Csqrt%7Bs_1%5E2%2Bs_2%5E2%7D%20%3D%20%5Csqrt%7B0.8%5E2%2B0.7875%5E2%7D%20%3D%201.1226)
What is the value of the test statistic?
![z = \frac{X - \mu}{s}](https://tex.z-dn.net/?f=z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7Bs%7D)
![z = \frac{2 - 0}{1.1226}](https://tex.z-dn.net/?f=z%20%3D%20%5Cfrac%7B2%20-%200%7D%7B1.1226%7D)
![z = 1.78](https://tex.z-dn.net/?f=z%20%3D%201.78)
The value of the test statistic is ![z = 1.78](https://tex.z-dn.net/?f=z%20%3D%201.78)