An oceanographer wants to test, on the basis of a random sample size of 20, whether the average depth of the ocean in certain areas is greater than 75 fathoms. At the .10 level of significance, what will the oceanographer decide if she observes a sample mean of 75.5? Assume that the population standard deviation is 1.75. (Also, compute the p-value)

Solution

Given that $n = 20$, sample mean $\overline{x} = 75$, the population standard deviation is $\sigma = 1.75$.

Step 1 Hypothesis Testing Problem

The hypothesis testing problem is
$H_0 : \mu = 75.5$ against $H_1 : \mu > 75.5$ ($\text{right-tailed}$)

Step 2 Test Statistic

The test statistic for testing above hypothesis testing problem is

$$ \begin{aligned} Z& =\frac{\overline{x} -\mu}{\sigma/\sqrt{n}}. \end{aligned} $$
The test statistic $Z$ follows $N(0,1)$ distribution.

Step 3 Significance Level

The significance level is $\alpha = 0.1$.

Step 4 Critical Value(s)

As the alternative hypothesis is $\text{right-tailed}$, the critical value of $Z$ $\text{is}$ $1.28$ (from Normal Statistical Table).

right-tailed z-critical region
right-tailed z-critical region

The rejection region (i.e. critical region) is $\text{Z > 1.28}$.

Step 5 Computation

The test statistic under the null hypothesis is

$$ \begin{aligned} Z_{obs}&=\frac{ \overline{x} -\mu_0}{\sigma/\sqrt{n}}\\ &= \frac{75-75.5}{1.75/ \sqrt{20 }}\\ &= -1.278 \end{aligned} $$

Step 6 Decision

Traditional Approach:

The test statistic is $Z_{obs} =-1.278$ which falls $\text{outside}$ the critical region, we $\text{fail to reject}$ the null hypothesis at $\alpha = 0.1$ level of significance.

Step 6 Decision

$p$-value Approach:

This is a $\text{right-tailed}$ test, so the p-value is the area to the $\text{right}$ of the test statistic ($Z_{obs}=-1.278$) is p-value = $0.8993$.

The p-value is $0.8993$ which is $\text{greater than}$ the significance level of $\alpha = 0.1$, we $\text{fail to reject}$ the null hypothesis at $\alpha =0.1$ level of significance.

There is no sufficient evidence to conclude that the average depth of the ocean in certain areas is greater than 75 fathoms.