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Chapter 8: Problem 7

For a Student's \(t\) distribution with \(d . f .=10\) and \(t=2.930\), (a) find an interval containing the corresponding \(P\) -value for a two-tailed test. (b) find an interval containing the corresponding \(P\) -value for a right- tailed test.

Short Answer

Expert verified
(a) Two-tailed P-value: between 0.02 and 0.01. (b) Right-tailed P-value: between 0.01 and 0.005.

Step by step solution

01

Identify the Degrees of Freedom

To find the intervals containing the P-value, first note that the degrees of freedom (d.f.) given is \(d.f. = 10\). This will help us determine the critical values of the \(t\)-distribution.
02

Two-tailed Test P-value

For a two-tailed test with \(t = 2.930\) and \(d.f. = 10\), we look at the \(t\)-distribution table or use statistical software. For \(t = 2.930\), the two-tailed P-value is twice the area to the right in one tail. With \(d.f. = 10\), a \(t\) value of 2.930 typically results in a two-tailed P-value between 0.02 and 0.01, based on common critical \(t\)-distribution ranges.
03

Right-tailed Test P-value

For a right-tailed test with the same \(t = 2.930\), we only consider one tail. This P-value is the area to the right of \(t = 2.930\) in the \(t\)-distribution with \(d.f. = 10\). Typically, this area corresponds to a P-value between 0.01 and 0.005.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Degrees of Freedom
Degrees of freedom, often abbreviated as d.f., play a crucial role when working with different types of data distributions, particularly the Student's t-distribution. In statistics, degrees of freedom refer to the number of independent values or quantities which can vary in the analysis without affecting the outcome. They are critical because they affect the critical values of statistical tests and thus the shape of the t-distribution.

For example, in a simple group comparison using the t-distribution, the degrees of freedom are typically calculated as the total number of observations minus the number of independent sample groups. This, in turn, determines how the tails of the t-distribution appear and how p-values are derived.

  • More degrees of freedom typically mean that the t-distribution approaches a normal distribution.
  • Fewer degrees of freedom imply a distribution with thicker tails, leading to larger critical t-values for hypothesis tests.
These adjustments help statisticians account for sample size variability and ensure accurate statistical inference.
Exploring Two-Tailed Tests
A two-tailed test is a type of hypothesis test used to discern if there is a significant difference in any direction between the expected and observed results.

This means a two-tailed test is employed when we are interested in deviations in both directions - both higher and lower than the expected mean. This kind of test is suitable when we do not have a directional hypothesis, making it ideal for tests where any divergence is of interest. The resulting P-value indicates the probability of observing the data, or something more extreme, in either of the direction tails of the distribution.

  • The resulting P-value is calculated by considering the extreme values from both ends of the distribution.
  • This doubles the area in one tail, explaining why the P-value appears larger compared to one-tailed tests.
  • Two-tailed tests are conservative, making it often harder to prove significance since deviations in both directions are considered.
Unpacking Right-Tailed Tests
A right-tailed test, or upper-tailed test, is a hypothesis test that focuses on discerning whether the observed statistic is significantly greater than expected. This type of test is used when our alternative hypothesis is directional; that is, we are only interested in the data values trending above a certain threshold.

In practice, for a right-tailed test, we compute the P-value by determining the probability of the test statistic falling to the right of our observed value (in technical terms, having a larger value). This confirms if the data provides strong evidence against the null hypothesis in the positive direction.

  • The right-tailed test emphasizes deviations on the higher side of the distribution.
  • It is particularly used in scenarios like testing if a process is improving significantly or if a new medication increases effects beyond the expected level.
  • The P-value for a right-tailed test is straightforwardly from one tail, making it simpler to calculate compared to two-tailed tests.

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Most popular questions from this chapter

Snow avalanches can be a real problem for travelers in the western United States and Canada. A very common type of avalanche is called the slab avalanche. These have been studied extensively by David McClung, a professor of civil engineering at the University of British Columbia. Slab avalanches studied in Canada have an average thickness of \(\mu=67\) (Source: Avalanche Handbook by D. McClung and P. Schaerer). The ski patrol at Vail, Colorado, is studying slab avalanches in its region. A random sample of avalanches in spring gave the following thicknesses (in \(\mathrm{cm})\) : \(\begin{array}{llllllll}59 & 51 & 76 & 38 & 65 & 54 & 49 & 62 \\ 68 & 55 & 64 & 67 & 63 & 74 & 65 & 79\end{array}\) i. Use a calculator with mean and standard deviation keys to verify that \(\bar{x}=61.8\) and \(s=10.6 \mathrm{~cm} .\) ii. Assume the slab thickness has an approximately normal distribution. Use a \(1 \%\) level of significance to test the claim that the mean slab thickness in the Vail region is different from that in Canada.

(a) For the same data and null hypothesis, is the \(P\) -value of a one-tailed test (right or left) larger or smaller than that of a two-tailed test? Explain. (b) For the same data, null hypothesis, and level of significance, is it possible that a one-tailed test results in the conclusion to reject \(H_{0}\) while a two-tailed test results in the conclusion to fail to reject \(H_{0}\) ? Explain. (c) For the same data, null hypothesis, and level of significance, if the conclusion is to reject \(H_{0}\) based on a two-tailed test, do you also reject \(H_{0}\) based on a one-tailed test? Explain. (d) If a report states that certain data were used to reject a given hypothesis, would it be a good idea to know what type of test (one-tailed or two-tailed) was used? Explain.

The western United States has a number of four-lane interstate highways that cut through long tracts of wilderness. To prevent car accidents with wild animals, the highways are bordered on both sides with 12 -foot-high woven wire fences. Although the fences prevent accidents, they also disturb the winter migration pattern of many animals. To compensate for this disturbance, the highways have frequent wilderness underpasses designed for exclusive use by deer, elk, and other animals. In Colorado, there is a large group of deer that spend their summer months in a region on one side of a highway and survive the winter months in a lower region on the other side. To determine if the highway has disturbed deer migration to the winter feeding area, the following data were gathered on a random sample of 10 wilderness districts in the winter feeding area. Row \(B\) represents the average January deer count for a 5 -year period before the highway was built, and row \(A\) represents the average January deer count for a 5 -year period after the highway was built. The highway department claims that the January population has not changed. Test this claim against the claim that the January population has dropped. Use a \(5 \%\) level of significance. Units used in the table are hundreds of deer. \begin{tabular}{l|cccccccccc} \hline Wilderness District & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline\(B:\) Before highway & \(10.3\) & \(7.2\) & \(12.9\) & \(5.8\) & \(17.4\) & \(9.9\) & \(20.5\) & \(16.2\) & \(18.9\) & \(11.6\) \\ \hline\(A:\) After highway & \(9.1\) & \(8.4\) & \(10.0\) & \(4.1\) & \(4.0\) & \(7.1\) & \(15.2\) & \(8.3\) & \(12.2\) & \(7.3\) \\ \hline \end{tabular}

Weatherwise magazine is published in association with the American Meteorological Society. Volume 46 , Number 6 has a rating system to classify Nor'easter storms that frequently hit New England states and can cause much damage near the ocean coast. A severe storm has an average peak wave height of \(16.4\) feet for waves hitting the shore. Suppose that a Nor'easter is in progress at the severe storm class rating. (a) Let us say that we want to set up a statistical test to see if the wave action (i.e., height) is dying down or getting worse. What would be the null hypothesis regarding average wave height? (b) If you wanted to test the hypothesis that the storm is getting worse, what would you use for the alternate hypothesis? (c) If you wanted to test the hypothesis that the waves are dying down, what would you use for the alternate hypothesis? (d) Suppose you do not know whether the storm is getting worse or dying out. You just want to test the hypothesis that the average wave height is different (either higher or lower) from the severe storm class rating. What would you use for the alternate hypothesis? (e) For each of the tests in parts (b), (c), and (d), would the area corresponding to the \(P\) -value be on the left, on the right, or on both sides of the mean? Explain your answer in each case.

Consider a test for \(\mu\). If the \(P\) -value is such that you can reject \(H_{0}\) for \(\alpha=0.01\), can you always reject \(H_{0}\) for \(\alpha=0.05\) ? Explain.
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