Question

question 1 of 8, step 2 of 3
correct
insurance company a claims that its customers pay less for car insurance, on average, than customers of its competitor, company b. you wonder if this is true, so you decide to compare the average monthly costs of similar insurance policies from the two companies. for a random sample of 12 people who buy insurance from company a, the mean cost is $151 per month with a standard deviation of $14. for 9 randomly selected customers of company b, you find that they pay a mean of $159 per month with a standard deviation of $12. assume that both populations are approximately normal and that the population variances are equal to test company as claim at the 0.05 level of significance. let customers of company a be population 1 and let customers of company b be population 2.
step 2 of 3: compute the value of the test statistic. round your answer to three decimal places.
answer

Explanation:

Step1: Calculate pooled variance

The formula for pooled variance $s_p^2$ is $s_p^2=\frac{(n_1 - 1)s_1^2+(n_2 - 1)s_2^2}{n_1 + n_2-2}$. Here, $n_1 = 12$, $s_1 = 14$, $n_2=9$, $s_2 = 12$.
\[

$$\begin{align*} s_p^2&=\frac{(12 - 1)\times14^2+(9 - 1)\times12^2}{12 + 9-2}\\ &=\frac{11\times196+8\times144}{19}\\ &=\frac{2156+1152}{19}\\ &=\frac{3308}{19}\\ &\approx174.105 \end{align*}$$

\]

Step2: Calculate the test - statistic

The formula for the two - sample t - test statistic when population variances are equal is $t=\frac{(\bar{x}_1-\bar{x}_2)-(\mu_1-\mu_2)}{s_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}$. Since we are testing if $\mu_1<\mu_2$, the null hypothesis is $H_0:\mu_1-\mu_2 = 0$ and the alternative hypothesis is $H_1:\mu_1-\mu_2<0$. Here, $\bar{x}_1 = 151$, $\bar{x}_2 = 159$, $\mu_1-\mu_2 = 0$, $s_p=\sqrt{174.105}\approx13.195$, $n_1 = 12$, $n_2 = 9$.
\[

$$\begin{align*} t&=\frac{(151 - 159)-0}{13.195\sqrt{\frac{1}{12}+\frac{1}{9}}}\\ &=\frac{- 8}{13.195\sqrt{\frac{3 + 4}{36}}}\\ &=\frac{-8}{13.195\sqrt{\frac{7}{36}}}\\ &=\frac{-8}{13.195\times\frac{\sqrt{7}}{6}}\\ &=\frac{-8}{13.195\times0.440}\\ &=\frac{-8}{5.806}\\ &\approx - 1.378 \end{align*}$$

\]

Answer:

$-1.378$