Question

you have been assigned to test the hypothesis that the average number of hours per week that an american works is higher than the average number of hours per week that a swedish works. the given data summarizes the sample statistics for the number of hours worked per week for workers in each country. assume that the population variances are unequal.

population 1 is defined as american workers and population 2 is defined as swedish workers.

a) what type of test should be performed based on the information provided? choose the correct answer below

a. non - pooled t
b. z
c. pooled t

b) what is the test statistic for this hypothesis test? round to two decimal places as needed.

test statistic =

Explanation:

Part (a)

To determine the test type, we check if population variances are equal (pooled) or not (non - pooled). The problem states population variances are unequal. For two - sample t - tests, when variances are unequal, we use a non - pooled t - test. A z - test is used when population standard deviations are known (not the case here with sample sds). A pooled t - test is for equal variances. So the correct test is non - pooled t.

Step 1: Recall the formula for the test statistic in a non - pooled two - sample t - test

The formula for the test statistic \(t\) is:

$$t=\frac{(\bar{x}_1-\bar{x}_2)-(\mu_1 - \mu_2)}{\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}}$$

Here, \(\mu_1-\mu_2 = 0\) (since we are testing if \(\mu_1>\mu_2\), the null hypothesis is \(\mu_1=\mu_2\)), \(\bar{x}_1 = 32.7\), \(\bar{x}_2=28.2\), \(s_1 = 5.1\), \(s_2 = 6.0\), \(n_1 = 15\), \(n_2=12\)

Step 2: Calculate the numerator

The numerator is \((\bar{x}_1-\bar{x}_2)-(\mu_1 - \mu_2)=(32.7 - 28.2)-0=4.5\)

Step 3: Calculate the denominator

First, calculate \(\frac{s_1^2}{n_1}\) and \(\frac{s_2^2}{n_2}\)
\(\frac{s_1^2}{n_1}=\frac{5.1^2}{15}=\frac{26.01}{15}=1.734\)
\(\frac{s_2^2}{n_2}=\frac{6.0^2}{12}=\frac{36}{12} = 3\)
Then, \(\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}=\sqrt{1.734 + 3}=\sqrt{4.734}\approx2.176\)

Step 4: Calculate the test statistic

\(t=\frac{4.5}{2.176}\approx2.07\)

Answer:

A. Non - pooled t

Part (b)