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Chapter 8: Problem 8

For a Student's \(t\) distribution with \(d . f .=16\) and \(t=-1.830\), (a) find an interval containing the corresponding \(P\) -value for a two-tailed test. (b) find an interval containing the corresponding \(P\) -vaiue for a left-tailed test.

Short Answer

Expert verified
Two-tailed p-value: between 0.05 and 0.10; left-tailed p-value: between 0.025 and 0.05.

Step by step solution

01

Understanding the Question

We are asked to find the interval that contains the p-value for both a two-tailed and a left-tailed test using Student's t-distribution with 16 degrees of freedom and a given t-value of -1.830.
02

Use Table for Two-tailed Test

To find the interval for a two-tailed test, we look up the critical values for the t-distribution table with 16 degrees of freedom. We find two t-values that the value -1.830 lies between and their corresponding two-tailed p-values.
03

Identify the Interval for Two-tailed Test

For a t-value of -1.830 with 16 degrees of freedom, using the t-table, the two-tailed p-value will lie between the p-values for t-values just larger and smaller than -1.830. Typically, this interval might be for t-values of -1.746 and -2.120, corresponding to approximate p-values of 0.05 and 0.10.
04

Use Table for Left-tailed Test

Now, for a left-tailed test, we again use the t-table. This time we find the left-tailed p-values corresponding to t-values around -1.830 with 16 degrees of freedom.
05

Identify the Interval for Left-tailed Test

For a t-value of -1.830, the left-tailed p-value lies between the values in the table for adjacent t-values. For example, if the p-value for t=-1.746 is 0.05 and for t=-2.120 is 0.025, the interval for -1.830 will fall between these two p-values.

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Key Concepts

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

Degrees of Freedom
Degrees of freedom (often abbreviated as df) is a crucial concept in statistics, particularly when working with the Student's t-distribution. It refers to the number of independent values or quantities that can vary in an analysis without breaking any constraints. In the context of a t-distribution, it is directly related to the sample size. In simple terms, the degrees of freedom are usually one less than the number of observations in the sample.
  • For example, if you have a sample size of 17, the degrees of freedom is 16 because df = n - 1, where n is the number of observations.
  • This concept plays a pivotal role in determining the exact shape of the t-distribution curve, as well as in calculations for determining p-values.
As the degrees of freedom increase, the t-distribution becomes more similar to a standard normal distribution. This is because with more data, our estimates become more precise.
P-value
The p-value is a powerful statistical measure that helps you determine the significance of your test results. It tells you the probability of observing your data, or something more extreme, assuming that the null hypothesis is true. In essence, a low p-value (typically 鈮� 0.05) indicates strong evidence against the null hypothesis, suggesting it should be rejected. On the other hand, a high p-value suggests that the observed data is not sufficiently unusual to reject the null hypothesis.
  • P-values are calculated differently based on the type of test (e.g., one-tailed or two-tailed).
  • They do not tell you the probability that the null hypothesis is true, just the probability of observing what you observed under the assumption that it's true.
Understanding p-values is essential as they allow researchers to gauge the strength of their findings.
Two-tailed Test
A two-tailed test is a statistical test used when you are interested in deviations on both sides of the assumed parameter. It is often employed when testing hypothesis where deviations could occur in either direction.
  • For instance, if you're testing whether a new drug is different from the existing one, you are not just interested in whether it is better but also if it could be worse.
  • In this test, the alternative hypothesis is not directional, meaning it could indicate a change, either higher or lower.
When you calculate a p-value for a two-tailed test, you double the area of one tail of the distribution because you're considering the extremities in both directions. This method ensures you're capturing the full range of possible deviations from the null hypothesis.
Left-tailed Test
A left-tailed test is one where the critical region or the region of rejection is in the left tail of the probability distribution. This type of test is used when you are interested in finding out if the observed value is significantly less than the assumed parameter.
  • An example scenario could involve testing if the average score of a test is less than a known or specified value.
  • The alternative hypothesis in a left-tailed test is directional, asserting that the parameter is less than a certain value.
Credit to the rejection region being solely in the left tail, you only consider the p-value that corresponds to the most extreme leftward outcomes for the test statistic, based on your degrees of freedom.

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Most popular questions from this chapter

Two populations have normal distributions. The first has population standard deviation 2 and the second has population standard deviation 3. A random sample of 16 measurements from the first population had a sample mean of \(20 .\) An independent random sample of 9 measurements from the second population had a sample mean of \(19 .\) Test the claim that the population mean of the first population exceeds that of the second. Use a \(5 \%\) level of significance. (a) What distribution does the sample test statistic follow? Explain. (b) State the hypotheses. (c) Compute \(\bar{x}_{1}-\bar{x}_{2}\) and the corresponding sample test statistic. (d) Find the \(P\) -value of the sample test statistic. (e) Conclude the test (f) The results.

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.

USA Today reported that about \(47 \%\) of the general consumer population in the United States is loyal to the automobile manufacturer of their choice. Suppose Chevrolet did a study of a random sample of 1006 Chevrolet owners and found that 490 said they would buy another Chevrolet. Does this indicate that the population proportion of consumers loyal to Chevrolet is more than \(47 \%\) ? Use \(\alpha=0.01\).

Is the national crime rate really going down? Some sociologists say yes! They say that the reason for the decline in crime rates in the \(1980 \mathrm{~s}\) and 1990 s is demographics. It seems that the population is aging, and older people commit fewer crimes. According to the FBI and the Justice Department, \(70 \%\) of all arrests are of males aged 15 to 34 years (Source: True Odds by \(\mathrm{J}\). Walsh, Merritt Publishing). Suppose you are a sociologist in Rock Springs, Wyoming, and a random sample of police files showed that of 32 arrests last month, 24 were of males aged 15 to 34 years. Use a \(1 \%\) level of significance to test the claim that the population proportion of such arrests in Rock Springs is different from \(70 \%\).

Would you favor spending more federal tax money on the arts? This question was asked by a research group on behalf of The National Institute (Reference: Painting by Numbers, J. Wypijewski, University of California Press). Of a random sample of \(n_{1}=93\) politically conservative voters, \(r_{1}=21\) responded yes. Another random sample of \(n_{2}=83\) politically moderate voters showed that \(r_{2}=22\) responded yes. Does this information indicate that the population proportion of conservative voters inclined to spend more federal tax money on funding the arts is less than the proportion of moderate voters so inclined? Use \(\alpha=0.05\).
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