Question

Finding P-values. In Exercises 5–8, either use technology to find the P-value or use Table A-3 to find a range of values for the P-value7. Old Faithful. The claim is that for the duration times (sec) of eruptions of the Old Faithful geyser, the mean is $\mu =240$sec. The sample size is n = 6 and the test statistic is t = 1.340.

Short answer

The range of the P-value is greater than 0.20.

Step-by-step solution

Given information

The claim states that the mean duration time of eruptions is 240 sec. The sample is $n=6$, and the test statistic is $t=1.340$.

State the hypotheses

The claim has an equality statement that will be the null hypothesis, and the alternate hypothesis will be the opposite of this.

Thus, the hypotheses are as follows.

$$\begin{gathered} H_0:\mu =240\sec \\ {H}_1:\mu \ne 240\sec \end{gathered}$$

Here, $\mu$ is the population mean time (sec) of eruptions of the old faithful geyser.

State the test statistic

The formula for the t-statistic is given below.

$t=\frac{\overline{x}-\mu }{\frac{s}{\sqrt{n}}}$

Here,

$$\begin{gathered} \overline{x}:\text{sample}\text{mean} \\ s:\text{sample}\text{stadard}\text{deviation} \\ \mu :\text{population}\text{mean} \\ n:\text{sample}\text{size} \end{gathered}$$

State the decision rule for the test

The decision rule is stated below for the level of significance $\alpha$.

$\text{If}\text{P - value}<\alpha$, reject the null hypothesis

If $\text{P - value}>\alpha$, fail to reject the null hypothesis.

Find the P-value range

In the given problem, the test-statistic is 1.340. The sample size is , and the degree of freedom of the t-distribution is

$$\begin{gathered} df=n-1 \\ =6-1 \\ =5 \end{gathered}$$

In the t-distribution table (Table A-3), look for the range where t statistic lies.

In the table, look for the closest bounds of the test statistic value in the row with a degree of freedom 5 for the two-tailed test.

In row 5, the test statistic value 1.340 is less than 1.476 (corresponding to 0.20 level for the two-tailed test).

Thus, the P-value will be greater than 0.20 and hence expressed as $\text{P - value}>0.20$.