Chapter 9.2, Problem 22P

Please provide the following information for Problems 11-22.

(a) What is the level of significance? State the null and alternate hypotheses.

(b)Check Requirements What sampling distribution will you use? Explain the rationale for your choice of sampling distribution. Compute the appropriate sampling distribution value of the sample test statistic.

(c) Find (or estimate) the P-value. Sketch the sampling distribution and show the area corresponding to the P- value.

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level ? $\alpha$

(e)Interpret your conclusion in the context of the application.

Note: For degrees of freedom d.f. not given in the Student's t table, use the closest d.f. that is smaller. In some situations, this choice of d.f. may increase the P-value by a small amount and therefore produce a slightly more “conservative" answer.

Archaeology: Tree Rings Tree-ring dating from archaeological excavation sites is used in conjunction with other chronologic evidence to estimate occupation dates of prehistoric Indian ruins in the southwestern United States. It is thought that Burnt Mesa Pueblo was occupied around 1300 A.D. (based on evidence from potsherds and stone tools). The following data give tree-ring dates (a.d.) from adjacent archaeological sites (Bandelier Archaeological Excavation Project: Summer 1990 Excavations at Burnt Mesa Pueblo, edited by T. Kohler, Washington State University Department of Anthropology):

1189	1267		1268	1275	1275
1271	1272		1316	1317	1230

Use a calculator with mean and standard deviation keys to verify that

$x\text{ = 1268 and }s\approx \text{ 37}\text{.29 years}\text{.}$

ii.Assuming the tree-ring dates in this excavation area follow a distribution that is approximately normal, does this information indicate that the population mean of tree-ring dates in the area is different from (cither higher or lower than) that in 1300 a.d.? Use a 1% level of significance.

To determine

Whether the sample mean $\overline{x}=1268$ and sample standard deviation $s=37.29$.

Answer to Problem 22P

Solution: Yes, the sample mean $\overline{x}=1268$ and sample standard deviation $s=37.29$.

Explanation of Solution

To calculate the required statistics using the Minitab, follow the below instructions:

Step 1: Go to the Minitab software.

Step 2: Go to Stat > Basic statistics > Display Descriptive Statistics.

Step 3: Select ‘Tree ring dates’ in variables.

Step 4: Click on OK.

The obtained statistics is:

Descriptive Statistics: Tree ring dates

Statistics

Variable	N	N*	Mean	SE Mean	StDev	Minimum	Q1	Median	Q3	Maximum
Tree ring dates	10	0	1268.0	11.8	37.3	1189.0	1257.8	1271.5	1285.3	1317.0

From the Minitab output, the sample mean and sample standard deviation are approximately equals to $\overline{x}=1268$ and $s=37.29$.

To determine

The level of significance, null and alternative hypothesis.

Answer to Problem 22P

Solution: The level of significance is $\alpha =0.01$. The null hypothesis is $H_0:\mu =1300$ and alternative hypothesis $H_A:\mu \ne 1300$.

Explanation of Solution

The level of significance is defined as the probability of rejecting the null hypothesis when it is true, it is denoted by $\alpha =0.01$.

Null hypothesis $H_0:\mu =1300$

Alternative hypothesis $H_A:\mu \ne 1300$

To determine

To find: The sampling distribution that should be used and compute the value of the sample test statistic.

Answer to Problem 22P

Solution: The student’s t distribution should be used. The sample test statistic is -2.71.

Explanation of Solution

Calculation:

We assume that x distribution is mound shape and symmetrical, because $\sigma$ is unknown, we can use student’s t distribution with $d.f=n-1=9$.

Using $\overline{x}=1268,\mu =1300,s=37.29,n=10$

The sample test statistic t is

$\begin{array}{l}t=\frac{(\overline{x}-\mu )}{\frac{s}{\sqrt{n}}}\\ t=\frac{(1268-1300)}{\frac{37.29}{\sqrt{10}}}\\ t=-2.7137\\ t\approx -2.71\end{array}$

To determine

To find: The P-value of the test statistic and sketch the sampling distribution showing the area corresponding to the P-value.

Answer to Problem 22P

Solution: The P-value of the test statistic is 0.024.

Explanation of Solution

Calculation:

We have t = -2.71

$\begin{array}{l}P-\text{value}\text{=}P(t<-2.71)+P(t>2.71)\\ P-\text{value}\text{=}2P(t<-2.71)\end{array}$

Using Table 4 from the Appendix to find the specified area:

$0.020<P-\text{value}<0.050$

Graph:

To draw the required graphs using the Minitab, follow the below instructions:

Step 1: Go to the Minitab software.

Step 2: Go to Graph > Probability distribution plot > View probability.

Step 3: Select ‘t’ and enter d.f = 9.

Step 4: Click on the Shaded area >X value.

Step 5: Enter X-value as -2.71 and select ‘Both Tail’.

Step 6: Click on OK.

The obtained distribution graph is:

$\begin{array}{l}P-\text{value}\text{=}2(0.012)\\ P-\text{value}\text{= }0.024\end{array}$

To determine

Whether we reject or fail to reject the null hypothesis and whether the data is statistically significant for a level of significance of 0.01.

Answer to Problem 22P

Solution: The P-value $>\alpha$, hence we fail to reject the $H_0$. The data is not statistically significant for a level of significance of 0.01.

Explanation of Solution

The P-value of 0.024 is greater than the level of significance ($\alpha$

) of 0.01. Therefore we don’t have enough evidence to reject the null hypothesis $H_0$, hence the data is not statistically significant for a level of significance of 0.01.

To determine

The interpretation for the conclusion.

Answer to Problem 22P

Solution: There is not enough evidence to conclude that population mean of tree ring dates in the area is different from that in 1300 A.D.

Explanation of Solution

The P-value of 0.024 is greater than the level of significance ($\alpha$

) of 0.01. Therefore we don’t have enough evidence to reject the null hypothesis $H_0$. There is not enough evidence to conclude that population mean of tree ring dates in the area is different from that in 1300 A.D.