Oxygen levels are measured to control breathing in symptoms of a disease. In the measurements taken from 242 people, it was observed that the oxygen values were normally distributed with an average of 33.3 and a standard deviation of 12.14. Are the oxygen levels of the patients above 30? Test with 0.01 significance level


Given that $n = 242$, sample mean $\overline{x} = 33.3$, the sample standard deviation is $\sigma = 12.14$.

Step 1 Hypothesis Testing Problem

The hypothesis testing problem is
$H_0 : \mu = 30$ against $H_1 : \mu > 30$ ($\text{right-tailed}$)

Step 2 Test Statistic

The test statistic for testing above hypothesis testing problem is

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

Step 3 Significance Level

The significance level is $\alpha = 0.01$.

Step 4 Critical Value(s)

As the alternative hypothesis is $\text{right-tailed}$, the critical value of $Z$ $\text{is}$ $2.33$ (from Normal Statistical Table).

z-critical right-tailed 0.01
z-critical right-tailed 0.01

The rejection region (i.e. critical region) is $\text{Z > 2.33}$.

Step 5 Computation

The test statistic under the null hypothesis is

$$ \begin{aligned} Z_{obs}&=\frac{ \overline{x} -\mu_0}{s/\sqrt{n}}\\ &= \frac{33.3-30}{12.14/ \sqrt{242 }}\\ &= 4.229 \end{aligned} $$

Step 6 Decision

Traditional Approach:

The test statistic is $Z_{obs} =4.229$ which falls $\text{inside}$ the critical region, we $\text{reject}$ the null hypothesis at $\alpha = 0.01$ 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 ($Z_{obs}=4.229$) is p-value = $0$.

The p-value is $0$ which is $\text{less than}$ the significance level of $\alpha = 0.01$, we $\text{reject}$ the null hypothesis at $\alpha =0.01$ level of significance.

There is sufficient evidence to conclude the oxygen levels of the patients is above 30.

Further Reading