A medical doctor wishes to test the claim that the standard deviation of the systolic blood pressure of deep sea divers is greater than 450. To do so, she selected a random sample of 25 divers and found $s = 468$. Assuming that the systolic blood pressures of deep sea divers are normally distributed, if the doctor wanted to test her research hypothesis at the 0.01 level of significance, what is the critical value?

#### Solution

Given that the sample size $n = 25$ and sample standard deviation $s = 468$.

#### Step 1 Hypothesis Problem

The hypothesis testing problem is

`$H_0 : \sigma = 450$`

against `$H_1 : \sigma > 450$`

($\text{right-tailed}$)

#### Step 2 Test Statistic

The test statistic for testing above hypothesis testing problem is

` $$ \chi^2 =\frac{(n-1)s^2}{\sigma^2} $$ `

#### Step 3 Level of Significance

The significance level is `$\alpha = 0.01$`

.

#### Step 4 Critical Value

As the alternative hypothesis is **$\text{right-tailed}$**, the critical value of $\chi^2$ $\text{ is }$ $\text{42.98}$ (from $\chi^2$ statistical table).

The rejection region (i.e. critical region) is $\chi^2 > 42.98$.

#### Step 5 Test Statistic

The test statistic under the null hypothesis is

` $$ \begin{aligned} \chi^2 &=\frac{(n-1)s^2}{\sigma^2_0}\\ &= \frac{(25-1)*(468)^2}{(450)^2}\\ &= 25.958 \end{aligned} $$ `

#### Step 6 Decision

Traditional approach

The test statistic is $\chi^2 =25.958$ which falls $outside$ the critical region, we **$\text{fail to reject}$** the null hypothesis.

$p$-value Approach

This is a **$\text{right-tailed}$** test, so the p-value is the area to the left of the test statistic ($\chi^2=25.958$) is p-value = $0.3553$.

The p-value is $0.3553$ which is **$\text{greater than}$** the significance level of $\alpha = 0.01$, we **$\text{fail to reject}$** the null hypothesis.

There is no sufficient evidence to support the claim that the standard deviation of the systolic blood pressure of deep sea divers is greater than 450.

#### Further Reading

- Statistics
- Descriptive Statistics
- Probability Theory
- Probability Distribution
- Hypothesis Testing
- Confidence interval
- Sample size determination
- Non-parametric Tests
- Correlation Regression
- Statistics Calculators