Question

find the standardized test statistic to test the claim that $\mu_1 = \mu_2$. two samples ar population. the sample statistics are given below.\
$n_1 = 40, n_2 = 35, \bar{x}_1 = 8, \bar{x}_2 = 9, \sigma_1 = 2.5, \sigma_2 = 2.8$\
\
\bigcirc a. $-1.0$\
\bigcirc b. $-0.8$\
\bigcirc c. $-1.6$\
\bigcirc d. $-2.6$

Explanation:

Step1: Recall the formula for the standardized test statistic (z - statistic) for two - sample z - test

The formula for the z - statistic when testing the claim that \(\mu_1=\mu_2\) (assuming two independent samples from normal populations with known population standard deviations \(\sigma_1\) and \(\sigma_2\)) is:

$$z=\frac{(\bar{x}_1 - \bar{x}_2)-(\mu_1-\mu_2)}{\sqrt{\frac{\sigma_1^{2}}{n_1}+\frac{\sigma_2^{2}}{n_2}}}$$

Since we are testing the claim that \(\mu_1 = \mu_2\), then \(\mu_1-\mu_2 = 0\). So the formula simplifies to:

$$z=\frac{\bar{x}_1-\bar{x}_2}{\sqrt{\frac{\sigma_1^{2}}{n_1}+\frac{\sigma_2^{2}}{n_2}}}$$

Step2: Substitute the given values into the formula

We are given that \(n_1 = 40\), \(n_2=35\), \(\bar{x}_1 = 8\), \(\bar{x}_2 = 9\), \(\sigma_1 = 2.5\), \(\sigma_2=2.8\)

First, calculate the numerator: \(\bar{x}_1-\bar{x}_2=8 - 9=- 1\)

Then, calculate the denominator:

$$
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$$

Now, calculate the z - statistic:

$$z=\frac{-1}{0.6166}\approx - 1.6$$

Answer:

C. - 1.6