Question

a hollywood studio believes that a movie that is considered a drama will draw a larger crowd on average than a movie that is considered a comedy. to test this theory, the studio randomly selects several movies that are classified as dramas and several movies that are classified as comedies and determines the box office revenue for each movie. the results of the survey are as follows. do the data substantiate the studios belief that dramas will draw a larger crowd on average than comedies at $alpha = 0.01$? let dramas be population 1 and comedies be population 2. assume that the population variances are approximately equal.

box office revenues (millions of dollars)

	$n$	$\bar{x}$	$s$
comedy	13	140	20

step 2 of 3: compute the value of the test statistic. round your answer to three decimal places.

Explanation:

Step1: Identify the formula for two - sample t - test statistic

For two independent samples with equal variances, the test statistic $t$ is given by $t=\frac{(\bar{x}_1-\bar{x}_2)-(\mu_1 - \mu_2)}{s_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}$, where $\mu_1-\mu_2 = 0$ (under the null hypothesis), $\bar{x}_1,\bar{x}_2$ are sample means, $n_1,n_2$ are sample sizes and $s_p$ is the pooled - standard deviation. First, calculate the pooled - standard deviation $s_p=\sqrt{\frac{(n_1 - 1)s_1^2+(n_2 - 1)s_2^2}{n_1 + n_2-2}}$.

Step2: Calculate the pooled - standard deviation $s_p$

Given $n_1 = 15$, $\bar{x}_1=180$, $s_1 = 60$, $n_2=13$, $\bar{x}_2 = 140$, $s_2=20$.
\[

$$\begin{align*} s_p&=\sqrt{\frac{(n_1 - 1)s_1^2+(n_2 - 1)s_2^2}{n_1 + n_2-2}}\\ &=\sqrt{\frac{(15 - 1)\times60^2+(13 - 1)\times20^2}{15 + 13-2}}\\ &=\sqrt{\frac{14\times3600+12\times400}{26}}\\ &=\sqrt{\frac{50400 + 4800}{26}}\\ &=\sqrt{\frac{55200}{26}}\\ &\approx\sqrt{2123.077}\\ &\approx46.077 \end{align*}$$

\]

Step3: Calculate the test statistic $t$

\[

$$\begin{align*} t&=\frac{(\bar{x}_1-\bar{x}_2)-(\mu_1 - \mu_2)}{s_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}\\ &=\frac{(180 - 140)-0}{46.077\sqrt{\frac{1}{15}+\frac{1}{13}}}\\ &=\frac{40}{46.077\sqrt{\frac{13 + 15}{15\times13}}}\\ &=\frac{40}{46.077\sqrt{\frac{28}{195}}}\\ &=\frac{40}{46.077\times0.378}\\ &=\frac{40}{17.427}\\ &\approx2.295 \end{align*}$$

\]

Answer:

$2.295$