Question

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): $\begin{array}{lllll}1189 & 1267 & 1268 & 1275 & 1275\end{array}$ $\begin{array}{lllll}1271 & 1272 & 1316 & 1317 & 1230\end{array}$ i. Use a calculator with mean and standard deviation keys to verify that $\bar{x}=1268$ and $s \approx 37.29$ years. 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 (either higher or lower than ) that in 1300 A.D.? Use a $1 \%$ level of significance.

Name: Problem 22, Chapter 8: Understandable Statistics Concepts and Methods
Uploaded: 2021-07-19T14:46:28-08:00
Duration: 4 min 27 s
Channel: Carolyn Behr-Jerome

   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):
$\begin{array}{lllll}1189 & 1267 & 1268 & 1275 & 1275\end{array}$
$\begin{array}{lllll}1271 & 1272 & 1316 & 1317 & 1230\end{array}$
i. Use a calculator with mean and standard deviation keys to verify that $\bar{x}=1268$ and $s \approx 37.29$ years.
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 (either higher or lower than ) that in 1300 A.D.? Use a $1 \%$ level of significance.

Understandable Statistics Concepts and Methods

Charles Brase,… 12th Edition

Chapter 8, Problem 22

Instant Answer

Solved by Expert Carolyn Behr-Jerome

07/19/2021

Step 1

We can do this by using a calculator with mean and standard deviation keys. After entering the given data, we find that the mean $\bar{x}$ is indeed 1268 and the standard deviation $s$ is approximately 37.29 years. Show more…

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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): $\begin{array}{lllll}1189 & 1267 & 1268 & 1275 & 1275\end{array}$ $\begin{array}{lllll}1271 & 1272 & 1316 & 1317 & 1230\end{array}$ i. Use a calculator with mean and standard deviation keys to verify that $\bar{x}=1268$ and $s \approx 37.29$ years. 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 (either higher or lower than ) that in 1300 A.D.? Use a $1 \%$ level of significance.

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Key Concepts

Level_of_Significance

The level of significance, denoted by alpha (?), is the threshold for determining when to reject the null hypothesis in a hypothesis test. It quantifies the probability of committing a Type I error, which is rejecting a true null hypothesis. This level is set before any analysis is conducted and helps to decide whether the observed results are statistically significant.

Null_and_Alternate_Hypotheses

The null hypothesis (H?) is a statement of no effect or no difference and is assumed true until evidence suggests otherwise. The alternate hypothesis (H? or Ha) states that there is an effect or a difference. In the context of testing a mean, the null hypothesis might state that the population mean equals a particular value, while the alternate hypothesis posits that the population mean is either not equal to or is different (greater or less) than that specific value.

Sampling_Distribution

The sampling distribution represents the probability distribution of a given statistic (like the sample mean) over many repeated samples. When the population standard deviation is unknown and the sample size is small, the Student's t-distribution is typically used because it accounts for the extra uncertainty. This distribution is approximately normal if the underlying data are normally distributed, which justifies using it in such tests.

Test_Statistic

The test statistic is a standardized value that measures how far the sample statistic is from the hypothesized population parameter, relative to the standard error of the statistic. In tests involving means with unknown population variance and smaller sample sizes, the t-statistic is calculated by subtracting the hypothesized mean from the sample mean and dividing by the estimate of the standard error, thereby allowing comparison with the t-distribution.

P_Value

The P-value represents the probability of obtaining a test statistic as extreme as, or more extreme than, the observed value under the assumption that the null hypothesis is true. It quantifies the evidence against the null hypothesis; a small P-value indicates strong evidence in favor of the alternative hypothesis. This value is then compared to the significance level to decide whether or not to reject the null hypothesis.

Decision_Rule

The decision rule in hypothesis testing involves comparing the P-value to the predetermined level of significance. If the P-value is less than or equal to ?, the null hypothesis is rejected in favor of the alternative. Conversely, if the P-value exceeds ?, there is insufficient evidence to reject the null hypothesis. This step formalizes the conclusion of the hypothesis test.

Degrees_of_Freedom

Degrees of freedom (df) pertain to the number of independent values in a calculation that are free to vary. In the context of the t-test, the degrees of freedom are typically calculated as the sample size minus one (n?1). The t-distribution’s shape depends on the degrees of freedom, and accurately determining df is crucial for selecting the correct critical value or P-value from the t-table during hypothesis testing.

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