In a pulse rate research, a simple random sample of 400 men results in a mean of 80 beats per minute, and a standard deviation of 10.8 beats per minute. Based on the sample results, the researcher concludes that the pulse rates of men have a standard deviation greater than 10 beats per minutes. Use a 0.05 significance level to test the researcher's claim.

(a) Identify the null hypothesis and alternative hypothesis.

(b) Determine the test statistic.

(c) Determine the P-value for this test.

(d) Is there sufficient evidence to support the researcher's claim? Explain

#### Solution

Given that the sample size `$n = 400$`

and sample standard deviation `$s = 10.8$`

.

(a) The null and alternative hypothesis are

`$H_0 : \sigma = 10$`

against `$H_1 : \sigma > 10$`

($\text{right-tailed}$)

(b) The test statistic for testing above hypothesis testing problem is

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

The test statistic under the null hypothesis is

` $$ \begin{aligned} \chi^2 &=\frac{(n-1)s^2}{\sigma^2_0}\\ & = \frac{(400-1)*(10.8)^2}{(100)}\\ &= 465.394 \end{aligned} $$ `

The test statistic $\chi^2$ follows $\chi^2$ distribution with $399$ degrees of freedom.

(c) The $p$-value for this test is the area to the right of the test statistic (`$\chi^2=465.394$`

) is p-value = `$P(\chi^2_{n-1}>465.394)= 0.0121$`

.

(d) The p-value is `$0.0121$`

which is **$\text{less than}$** the significance level of $\alpha = 0.05$, we **$\text{reject}$** the null hypothesis.

There is sufficient evidence to support the researcher's claim.

#### Further Reading

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