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


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.

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