Math / Data & Statistics

QuestionTo test $\mathrm{H}_{0} \cdot \mu=40$ versus $\mathrm{H}_{1} \cdot \mu<40$ , a random sample of size $n=25$ is obtained from a population that is known to be normally distributed. Complete parts (a) through (d) below. Click here to view the $t$ -Distribution Area in Right Tail (a) If $\bar{x}=37.5$ and $s=11.9$ , compute the test statistic, $\mathrm{t}_{0}=$ $\square$ (Round to three decimal places as needed.)

Studdy Solution

STEP 1

1. The null hypothesis is $H_0: \mu = 40$ .
2. The alternative hypothesis is $H_1: \mu < 40$ .
3. The sample mean is $\bar{x} = 37.5$ .
4. The sample standard deviation is $s = 11.9$ .
5. The sample size is $n = 25$ .
6. The population is normally distributed.

STEP 2

1. Calculate the standard error of the mean.
2. Compute the test statistic $t_0$ .

STEP 3

Calculate the standard error of the mean using the formula:
$SE = \frac{s}{\sqrt{n}}$
Substitute the given values:
$SE = \frac{11.9}{\sqrt{25}} = \frac{11.9}{5} = 2.38$

STEP 4

Compute the test statistic $t_0$ using the formula:
$t_0 = \frac{\bar{x} - \mu}{SE}$
Substitute the given values:
$t_0 = \frac{37.5 - 40}{2.38} = \frac{-2.5}{2.38} \approx -1.050$
The test statistic is:
$t_0 \approx \boxed{-1.050}$

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