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Luden [163]
3 years ago
5

Consider the following hypothesis test: H0: μ1 - μ2 = 0 Ha: μ1 - μ2 ≠ 0 There are two independent samples taken from the two pop

ulations: 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.
Mathematics
1 answer:
nlexa [21]3 years ago
5 0

Answer:

The value of the test statistic is z = 1.78

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 \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 = \frac{\sigma}{\sqrt{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 = \frac{7.2}{\sqrt{81}} = 0.8

Sample 2:

\mu_2 = 108, s_2 = \frac{6.3}{\sqrt{64}} = 0.7875

The test statistic is:

z = \frac{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 = \sqrt{s_1^2+s_2^2} = \sqrt{0.8^2+0.7875^2} = 1.1226

What is the value of the test statistic?

z = \frac{X - \mu}{s}

z = \frac{2 - 0}{1.1226}

z = 1.78

The value of the test statistic is z = 1.78

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