Question

Consider the following hypothesis test: H0: μ1 - μ2 = 0 Ha: μ1 - μ2 ≠ 0 There are two independent samples taken from the two populations: Sample 1 Sample 2 n1 = 81 n2 = 64 Sample average = 110 Sample average = 108 Population standard deviation = 7.2 Population standard deviation = 6.3 ​ What is the value of the test statistic.

Answer 1

The value of the test statistic is
`z = 1.78`

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
`\mu` and standard deviation
`\sigma`, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
`\mu` and standard deviation
`s = (\sigma)/(√(n))`.

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 = (7.2)/(√(81)) = 0.8`

Sample 2:

`\mu_2 = 108, s_2 = (6.3)/(√(64)) = 0.7875`

The test statistic is:

`z = (X - \mu)/(s)`

In which X is the sample mean,
`\mu` 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`

Distribution of the difference:

`X = \mu_1 - \mu_2 = 110 - 108 = 2`

`s = √(s_1^2+s_2^2) = √(0.8^2+0.7875^2) = 1.1226`

What is the value of the test statistic?

`z = (X - \mu)/(s)`

`z = (2 - 0)/(1.1226)`

`z = 1.78`

The value of the test statistic is
`z = 1.78`