Question

a manufacturer claims that the calling range (in feet) of its 900 - mhz cordless telephone is greater than that of its leading competitor. a sample of 5 phones from the manufacturer had a mean range of 1180 feet with a standard deviation of 38 feet. a sample of 12 similar phones from its competitor had a mean range of 1120 feet with a standard deviation of 27 feet. do the results support the manufacturers claim? let $mu_1$ be the true mean range of the manufacturers cordless telephone and $mu_2$ be the true mean range of the competitors cordless telephone. use a significance level of $alpha = 0.1$ for the test. assume that the population variances are equal and that the two populations are normally distributed. step 2 of 4: compute the value of the $t$ - test statistic. round your answer to three decimal places.

Explanation:

Step1: Calculate the pooled - variance

The formula for the pooled - variance $s_p^2$ when the population variances are equal is $s_p^2=\frac{(n_1 - 1)s_1^2+(n_2 - 1)s_2^2}{n_1 + n_2-2}$, where $n_1 = 5$, $s_1 = 38$, $n_2=12$, and $s_2 = 27$.
\[

$$\begin{align*} s_p^2&=\frac{(5 - 1)\times38^2+(12 - 1)\times27^2}{5 + 12-2}\\ &=\frac{4\times1444+11\times729}{15}\\ &=\frac{5776+8019}{15}\\ &=\frac{13795}{15}\approx919.667 \end{align*}$$

\]

Step2: Calculate the t - test statistic

The formula for the t - test statistic for two - sample independent t - test with equal variances 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 the null hypothesis is $H_0:\mu_1-\mu_2\leq0$ and the alternative hypothesis is $H_1:\mu_1 - \mu_2>0$, and we assume $\mu_1-\mu_2 = 0$ under the null hypothesis, $\bar{x}_1 = 1180$, $\bar{x}_2 = 1120$, $s_p=\sqrt{919.667}\approx30.326$, $n_1 = 5$, and $n_2 = 12$.
\[

$$\begin{align*} t&=\frac{(1180 - 1120)-0}{30.326\sqrt{\frac{1}{5}+\frac{1}{12}}}\\ &=\frac{60}{30.326\sqrt{\frac{12 + 5}{60}}}\\ &=\frac{60}{30.326\sqrt{\frac{17}{60}}}\\ &=\frac{60}{30.326\times0.531}\\ &=\frac{60}{16.104}\approx3.726 \end{align*}$$

\]

Answer:

$3.726$