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Please provide the following information for Problems. (a) What is the level of significance? State the null and alternate hypotheses. (b) Check Requirements What sampling distribution will you use? What assumptions are you making? Compute the sample test statistic and corresponding $z$ or $t$ value as appropriate. (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 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 a small amount and therefore produce a slightly more "conservative" answer. Wildlife: Fox Rabies A study of fox rabies in southern Germany gave the following information about different regions and the occurrence of rabies in each region (Reference: $\mathbf{B}$. Sayers et al., "A Pattern Analysis Study of a Wildlife Rabies Epizootic," Medical Informatics, Vol. 2, pp. 1 1-34). Based on information from this article, a random sample of $n_{1}=16$ locations in region I gave the following information about the number of cases of fox $$\begin{array}{rlrrrrrrr} x_{1}: \text { Region I data } & 1 & 8 & 8 & 8 & 7 & 8 & 8 & 1 \\ & 3 & 3 & 3 & 2 & 5 & 1 & 4 & 6 \end{array}$$ A second random sample of $n_{2}=15$ locations in region II gave the following information about the number of cases of fox rabies near that location. $$\begin{array}{lllllllll} x_{2} \text { : Region II data } & 1 & 1 & 3 & 1 & 4 & 8 & 5 & 4 \\ & 4 & 4 & 2 & 2 & 5 & 6 & 9 & \end{array}$$ i. Use a calculator with sample mean and sample standard deviation keys to verify that $\bar{x}_{1}=4.75$ with $s_{1} \approx 2.82$ in region I and $\bar{x}_{2} \approx 3.93$ with $s_{2} \approx 2.43$ in region II. ii. Does this information indicate that there is a difference (either way) in the mean number of cases of fox rabies between the two regions? Use a $5 \%$ level of significance. (Assume the distribution of rabies cases in both regions is mound-shaped and approximately normal.)

Name: Problem 23, Chapter 8: Understandable Statistics, Concepts and Methods
Uploaded: 2021-11-19T00:00:23-08:00
Duration: 3 min 57 s
Channel: Sheryl Ezze

   Please provide the following information for Problems.
(a) What is the level of significance? State the null and alternate hypotheses.
(b) Check Requirements What sampling distribution will you use? What assumptions are you making? Compute the sample test statistic and corresponding $z$ or $t$ value as appropriate.
(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 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 a small amount and therefore produce a slightly more "conservative" answer.
Wildlife: Fox Rabies A study of fox rabies in southern Germany gave the following information about different regions and the occurrence of rabies in each region (Reference: $\mathbf{B}$. Sayers et al., "A Pattern Analysis Study of a Wildlife Rabies Epizootic," Medical Informatics, Vol. 2, pp. 1 1-34). Based on information from this article, a random sample of $n_{1}=16$ locations in region I gave the following information about the number of cases of fox
$$\begin{array}{rlrrrrrrr}
x_{1}: \text { Region I data } & 1 & 8 & 8 & 8 & 7 & 8 & 8 & 1 \\
& 3 & 3 & 3 & 2 & 5 & 1 & 4 & 6
\end{array}$$
A second random sample of $n_{2}=15$ locations in region II gave the following information about the number of cases of fox rabies near that location.
$$\begin{array}{lllllllll}
x_{2} \text { : Region II data } & 1 & 1 & 3 & 1 & 4 & 8 & 5 & 4 \\
& 4 & 4 & 2 & 2 & 5 & 6 & 9 &
\end{array}$$
i. Use a calculator with sample mean and sample standard deviation keys
to verify that $\bar{x}_{1}=4.75$ with $s_{1} \approx 2.82$ in region I and $\bar{x}_{2} \approx 3.93$ with $s_{2} \approx 2.43$ in region II.
ii. Does this information indicate that there is a difference (either way) in the mean number of cases of fox rabies between the two regions? Use a
$5 \%$ level of significance. (Assume the distribution of rabies cases in both regions is mound-shaped and approximately normal.)

Understandable Statistics, Concepts and Methods

Charles Henry Brase,… 12th Edition

Chapter 8, Problem 23

Instant Answer

Solved by Expert Sheryl Ezze

11/19/2021

Step 1

The level of significance is $\alpha=0.05$. The null hypothesis $H_{0}$ is that the mean number of cases of fox rabies in the two regions are equal, i.e., $\mu_{1}=\mu_{2}$. The alternate hypothesis $H_{1}$ is that the mean number of cases of fox rabies in the two Show more…

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Please provide the following information for Problems. (a) What is the level of significance? State the null and alternate hypotheses. (b) Check Requirements What sampling distribution will you use? What assumptions are you making? Compute the sample test statistic and corresponding $z$ or $t$ value as appropriate. (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 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 a small amount and therefore produce a slightly more "conservative" answer. Wildlife: Fox Rabies A study of fox rabies in southern Germany gave the following information about different regions and the occurrence of rabies in each region (Reference: $\mathbf{B}$. Sayers et al., "A Pattern Analysis Study of a Wildlife Rabies Epizootic," Medical Informatics, Vol. 2, pp. 1 1-34). Based on information from this article, a random sample of $n_{1}=16$ locations in region I gave the following information about the number of cases of fox $$\begin{array}{rlrrrrrrr} x_{1}: \text { Region I data } & 1 & 8 & 8 & 8 & 7 & 8 & 8 & 1 \\ & 3 & 3 & 3 & 2 & 5 & 1 & 4 & 6 \end{array}$$ A second random sample of $n_{2}=15$ locations in region II gave the following information about the number of cases of fox rabies near that location. $$\begin{array}{lllllllll} x_{2} \text { : Region II data } & 1 & 1 & 3 & 1 & 4 & 8 & 5 & 4 \\ & 4 & 4 & 2 & 2 & 5 & 6 & 9 & \end{array}$$ i. Use a calculator with sample mean and sample standard deviation keys to verify that $\bar{x}_{1}=4.75$ with $s_{1} \approx 2.82$ in region I and $\bar{x}_{2} \approx 3.93$ with $s_{2} \approx 2.43$ in region II. ii. Does this information indicate that there is a difference (either way) in the mean number of cases of fox rabies between the two regions? Use a $5 \%$ level of significance. (Assume the distribution of rabies cases in both regions is mound-shaped and approximately normal.)

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Key Concepts

Interpretation of Results

Interpreting the results of a hypothesis test involves making a conclusion about the population based on the sample data. If the null hypothesis is rejected, it suggests that there is a statistically significant difference or effect, whereas if it is not rejected, the evidence is insufficient to support the alternative hypothesis. This interpretation should be made in the context of the research question and with consideration of the assumptions and limitations of the test.

P-Value and Decision Making

The p-value is the probability of obtaining an effect at least as extreme as that observed in the sample data, assuming that the null hypothesis is true. It is used to decide whether to reject the null hypothesis. If the p-value is less than or equal to the significance level, there is sufficient evidence to reject the null hypothesis, suggesting that the observed effect is statistically significant.

Test Statistic and Degrees of Freedom

The test statistic quantifies the degree of departure of the sample statistic from the null hypothesis, usually calculated as a standardized measure (such as a t-value or z-value). In the context of a two-sample t-test, degrees of freedom (df) are used to determine which t-distribution to reference. When the sample sizes or variances differ, degrees of freedom are often approximated using methods such as the Welch-Satterthwaite equation, and sometimes a more conservative choice (the smaller df) is made to ensure valid inferences.

Two-Sample t-Test

A two-sample t-test is used to determine whether there is a statistically significant difference between the means of two independent groups. It is particularly useful when comparing the means from small sample sizes or when the population standard deviations are not known. The test statistic is computed based on the difference between the sample means relative to the variability observed in the data, and it follows a t-distribution under the null hypothesis.

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence in a sample of data to infer that a certain condition holds for the entire population. It involves formulating a null hypothesis (typically representing no effect or no difference) and an alternative hypothesis (representing a claimed effect or difference), and then using sample data to decide which hypothesis is more plausible based on a significance level.

Level of Significance

The level of significance, often denoted by ?, is the threshold for rejecting the null hypothesis. It represents the probability of making a Type I error, which occurs when the null hypothesis is rejected even though it is true. Setting this level (e.g., 0.05 for a 5% significance level) guides the decision-making process by providing a clear cutoff for what constitutes statistically significant evidence against the null hypothesis.

Null and Alternative Hypotheses

In hypothesis testing, the null hypothesis (H?) is the default assumption that there is no effect or no difference between groups or parameters, while the alternative hypothesis (H?) contradicts the null hypothesis and represents what the analyst seeks evidence for. Clearly defining these hypotheses before the analysis is crucial to avoid bias and to ensure that the test’s conclusions are valid.

Assumptions and Sampling Distribution

Statistical tests, such as the t-test, rely on key assumptions about the underlying data, including that the observations are independent, the populations are approximately normally distributed, and, in some cases, that variances are equal. The choice of the sampling distribution (e.g., t-distribution versus z-distribution) depends on whether the population standard deviation is known and on the sample size. When these assumptions are met, the sampling distribution accurately reflects the behavior of the test statistic under the null hypothesis.

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