Question

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. Agriculture: Bell Peppers The pathogen Phytophthora capsici causes bell peppers to wilt and die. Because bell peppers are an important commercial crop, this disease has undergone a great deal of agricultural research. It is thought that too much water aids the spread of the pathogen. Two fields are under study. The first step in the research project is to compare the mean soil water content for the two fields (Source: Journal of Agricultural, Biological, and Environmental Statistics, Vol. $2,$ No. 2 ). Units are percent water by volume of soil. Field A samples, $x_{1}$ $\begin{array}{cccccc}10.2 & 10.7 & 15.5 & 10.4 & 9.9 & 10.0 & 16.6\end{array}$ $\begin{array}{cccccc}-15.1 & 15.2 & 13.8 & 14.1 & 11.4 & 11.5 & 11.0\end{array}$ i. Use a calculator with mean and standard deviation keys to verify that $\bar{x}_{1} \approx 12.53, s_{1} \approx 2.39, \bar{x}_{2} \approx 10.77,$ and $s_{2}=2.40$ ii. Assuming the distribution of soil water content in each field is moundshaped and symmetric, use a $5 \%$ level of significance to test the claim that field A has, on average, a higher soil water content than field B.

Name: Problem 24, Chapter 8: Understandable Statistics, Concepts and Methods
Uploaded: 2021-11-19T21:28:49-08:00
Duration: 3 min 11 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.
Agriculture: Bell Peppers The pathogen Phytophthora capsici causes bell peppers to wilt and die. Because bell peppers are an important commercial crop, this disease has undergone a great deal of agricultural research. It is thought that too much water aids the spread of the pathogen. Two fields are under study. The first step in the research project is to compare the mean soil water content for the two fields (Source: Journal of Agricultural, Biological, and Environmental Statistics, Vol. $2,$ No. 2 ). Units are percent water by volume of soil.
Field A samples, $x_{1}$
$\begin{array}{cccccc}10.2 & 10.7 & 15.5 & 10.4 & 9.9 & 10.0 & 16.6\end{array}$ 
$\begin{array}{cccccc}-15.1 & 15.2 & 13.8 & 14.1 & 11.4 & 11.5 & 11.0\end{array}$
i. Use a calculator with mean and standard deviation keys to verify that $\bar{x}_{1} \approx 12.53, s_{1} \approx 2.39, \bar{x}_{2} \approx 10.77,$ and $s_{2}=2.40$
ii. Assuming the distribution of soil water content in each field is moundshaped and symmetric, use a $5 \%$ level of significance to test the claim that field A has, on average, a higher soil water content than field B.

Understandable Statistics, Concepts and Methods

Charles Henry Brase,… 12th Edition

Chapter 8, Problem 24

Instant Answer

Solved by Expert Sheryl Ezze

11/19/2021

Step 1

The null hypothesis is that the mean soil water content of field A is equal to that of field B, while the alternate hypothesis is that the mean soil water content of field A is greater than that of field B. In mathematical terms, this can be written as: Null 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. Agriculture: Bell Peppers The pathogen Phytophthora capsici causes bell peppers to wilt and die. Because bell peppers are an important commercial crop, this disease has undergone a great deal of agricultural research. It is thought that too much water aids the spread of the pathogen. Two fields are under study. The first step in the research project is to compare the mean soil water content for the two fields (Source: Journal of Agricultural, Biological, and Environmental Statistics, Vol. $2,$ No. 2 ). Units are percent water by volume of soil. Field A samples, $x_{1}$ $\begin{array}{cccccc}10.2 & 10.7 & 15.5 & 10.4 & 9.9 & 10.0 & 16.6\end{array}$ $\begin{array}{cccccc}-15.1 & 15.2 & 13.8 & 14.1 & 11.4 & 11.5 & 11.0\end{array}$ i. Use a calculator with mean and standard deviation keys to verify that $\bar{x}_{1} \approx 12.53, s_{1} \approx 2.39, \bar{x}_{2} \approx 10.77,$ and $s_{2}=2.40$ ii. Assuming the distribution of soil water content in each field is moundshaped and symmetric, use a $5 \%$ level of significance to test the claim that field A has, on average, a higher soil water content than field B.

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Key Concepts

Contextual Interpretation

The interpretation of the test results should relate the statistical findings back to the real-world context from which the data were drawn. This involves explaining what rejecting or failing to reject the null hypothesis means in terms of the practical implications, such as the effectiveness of a treatment or the difference between two populations in a specific application. It is essential to communicate these conclusions in clear terms relating to the subject matter being studied.

Statistical Decision Making

Statistical decision making involves comparing the calculated P-value to the level of significance. If the P-value is less than ?, the null hypothesis is rejected, suggesting that the observed effect is statistically significant. If the P-value is greater than or equal to ?, there is insufficient evidence to reject the null hypothesis. This method ensures that decisions are based on the probability of observing the data under the null hypothesis.

P-Value

The P-value represents the probability of obtaining a test statistic at least as extreme as the one observed, assuming that the null hypothesis is true. A small P-value indicates that the observed data would be highly unlikely under the null hypothesis, thereby providing evidence against it. In hypothesis testing, the P-value is compared with the level of significance to decide whether to reject the null hypothesis.

Test Statistic

The test statistic quantifies how far the sample statistic is from the null hypothesis value in units of standard error. For a two-sample t-test, the statistic is calculated by finding the difference between the sample means, then dividing by the appropriate standard error, which is derived from the sample standard deviations and sample sizes. The resulting value is used to assess the probability of observing such a difference if the null hypothesis were true.

Level of Significance

The level of significance (often denoted by ?) is a threshold used to determine whether the observed data are sufficiently rare under the null hypothesis. It represents the probability of committing a Type I error, i.e., rejecting a true null hypothesis. Commonly used values include 0.05 and 0.01, and it sets the standard for how extreme the test statistic must be to favor the alternative hypothesis.

Sampling Distribution

The sampling distribution describes the probability distribution of a test statistic based on a random sample. For comparing two means with relatively small samples and an assumption of approximately symmetric distributions, the Student’s t-distribution is used. This distribution accounts for the additional variability introduced by estimating the population standard deviation from sample data.

Null and Alternate Hypotheses

The null hypothesis (H0) and alternate hypothesis (H1) are statements about the parameters of populations. In a two-sample test, the null hypothesis typically asserts that there is no difference (or that one mean is equal to or not greater than the other, depending on the test type), while the alternate hypothesis represents a claim of a difference (or a one-directional difference when a one-sided test is appropriate). Crafting these hypotheses correctly is fundamental to directing the analysis.

Hypothesis Testing

Hypothesis testing is a statistical procedure used to evaluate evidence provided by data against a stated hypothesis. In the context of comparing two populations, it involves formulating a null hypothesis (typically a claim of no difference or effect) and an alternative hypothesis (reflecting the anticipated difference or effect), then using sample data to decide whether the evidence is strong enough to reject the null hypothesis.

Assumptions

Key assumptions in a two-sample inference, such as the independent samples t-test, include independence of observations, roughly mound-shaped (or normally distributed) distributions, and, depending on the specific t-test, equal or unequal variances. These assumptions ensure that the test statistic follows the appropriate distribution (t-distribution) and that the conclusions drawn are valid.

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