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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

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(a) Two-tailed P-value interval: 0.02 to 0.04. (b) Right-tailed P-value interval: 0.01 to 0.02.

Step by step solution

01

Determine the Two-Tailed P-value Interval

First, for a two-tailed test with a given degree of freedom (d.f.) of 10 and a test statistic \(t = 2.930\), you need to find where this value falls within the t-distribution table. Look up the critical values for a two-tailed test with d.f. = 10. The critical values closest to \(t = 2.930\) typically correspond to \(t = 2.764\) for \(P = 0.02\) and \(t = 3.169\) for \(P = 0.01\). Thus, the two-tailed P-value interval for \(t = 2.930\) is between 0.02 and 0.04.
02

Determine the Right-Tailed P-value Interval

For a right-tailed test, use the same test statistic \(t = 2.930\) with d.f. = 10. You consider only one side of the t-distribution. Therefore, the P-value for the right-tailed test corresponds to half of the two-tailed P-value because we consider only one tail. For \(t = 2.930\), since the two-tailed P-value is between 0.02 and 0.04, the right-tailed P-value is between 0.01 and 0.02.

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Key Concepts

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

P-value
The P-value is a crucial concept in hypothesis testing. It helps you determine the significance of your test results. Essentially, the P-value is the probability of observing results at least as extreme as those measured, assuming the null hypothesis is true.

If the P-value is small, say less than 0.05, it indicates strong evidence against the null hypothesis. In such cases, you might consider rejecting the null hypothesis. Conversely, a large P-value suggests that you should not reject the null hypothesis.

Here are some key points about P-values:
  • P-values quantify the strength of the evidence against the null hypothesis.
  • They are used to make decisions in hypothesis testing, such as rejecting or failing to reject the null hypothesis.
  • P-values can be determined using statistical software or by looking them up in statistical tables specific to the distribution you are dealing with, such as the Student's t-distribution.
Understanding P-values helps you make informed decisions based on your data, offering a numeric way to interpret statistical results.
two-tailed test
A two-tailed test is a type of hypothesis test where the area in both tails of the distribution is considered. It is used when you're interested in finding out whether your observed data is significantly different from the expected data in either direction. This means both significantly higher or significantly lower.

In the context of the Student's t-distribution:
  • The two-tailed test checks the extremity of your test statistic in both tails of the distribution.
  • You'll look up the critical values in a t-distribution table that correspond to your test statistic.
  • The P-value from a two-tailed test is essentially twice that of a one-tailed test at the same test statistic.
A two-tailed test is appropriate when the research question seeks to determine any change or difference, without specifying a direction.
right-tailed test
In contrast to a two-tailed test, a right-tailed test focuses solely on one side of the distribution. Specifically, the right tail. This is used when you are only interested in determining if the observed data is significantly greater than the expected value.

Key aspects of a right-tailed test include:
  • The critical area, or the rejection zone, is located in the right tail of the distribution.
  • To determine the right-tailed P-value, you consider only the area in the right tail above your test statistic.
  • Because the focus is on one side, the P-value for a right-tailed test is half that of a two-tailed test with the same test statistic.
A right-tailed test is used when your hypothesis specifically predicts an increase or a value greater than the null hypothesis.

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Most popular questions from this chapter

Is there a relationship between confidence intervals and two-tailed hypothesis tests? Let \(c\) be the level of confidence used to construct a confidence interval from sample data. Let \(\alpha\) be the level of significance for a two-tailed hypothesis test. The following statement applies to hypothesis tests of the mean. For a two-tailed hypothesis test with level of significance \(\alpha\) and null hypothesis \(H_{0}\) : \(\mu=k,\) we reject \(H_{0}\) whenever \(k\) falls outside the \(c=1-\alpha\) confidence interval for \(\mu\) based on the sample data. When \(k\) falls within the \(c=1-\alpha\) confidence interval, we do not reject \(H_{0}\) (A corresponding relationship between confidence intervals and two-tailed hypothesis tests also is valid for other parameters, such as \(p, \mu_{1}-\mu_{2},\) and \(p_{1}-p_{2},\) which we will study in Sections 8.3 and \(8.5 .\) ) Whenever the value of \(k\) given in the null hypothesis falls outside the \(c=1-\alpha\) confidence interval for the parameter, we reject \(H_{0} .\) For example, consider a two-tailed hypothesis test with \(\alpha=0.01\) and $$H_{0}: \mu=20 \quad H_{1}: \mu \neq 20$$ A random sample of size 36 has a sample mean \(\bar{x}=22\) from a population with standard deviation \(\sigma=4.\) (a) What is the value of \(c=1-\alpha ?\) Using the methods of Chapter \(7,\) construct a \(1-\alpha\) confidence interval for \(\mu\) from the sample data. What is the value of \(\mu\) given in the null hypothesis (i.e., what is \(k\) )? Is this value in the confidence interval? Do we reject or fail to reject \(H_{0}\) based on this information? (b) Using methods of this chapter, find the \(P\) -value for the hypothesis test. Do we reject or fail to reject \(H_{0}\) ? Compare your result to that of part (a).

Suppose you want to test the claim that a population mean equals \(40 .\) (a) State the null hypothesis. (b) State the alternate hypothesis if you have no information regarding how the population mean might differ from \(40 .\) (c) State the alternate hypothesis if you believe (based on experience or past studies) that the population mean may exceed \(40 .\) (d) State the alternate hypothesis if you believe (based on experience or past studies) that the population mean may be less than \(40 .\)

Sociology: Crime Rate 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 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 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 \(70 \%.\)

Paired Differences Test For a random sample of 36 data pairs, the sample mean of the differences was 0.8. The sample standard deviation of the differences was \(2 .\) At the \(5 \%\) level of significance, test the claim that the population mean of the differences is different from 0. (a) Check Requirements Is it appropriate to use a Student's \(t\) distribution for the sample test statistic? Explain. What degrees of freedom are used? (b) State the hypotheses. (c) Compute the sample test statistic and corresponding \(t\) value. (d) Estimate the \(P\) -value of the sample test statistic. (e) Do we reject or fail to reject the null hypothesis? Explain. (f) Interpretation What do your results tell you?

What terminology do we use for the probability of rejecting the null hypothesis when it is, in fact, false?
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