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Chapter 10: Problem 2

(a) Determine the critical value for a right-tailed test of a population standard deviation with 16 degrees of freedom at the \(\alpha=0.01\) level of significance. (b) Determine the critical value for a left-tailed test of a population standard deviation for a sample of size \(n=14\) at the \(\alpha=0.01\) level of significance. (c) Determine the critical values for a two-tailed test of a population standard deviation for a sample of size \(n=61\) at the \(\alpha=0.05\) level of significance.

Short Answer

Expert verified
(a) 32.000. (b) 3.012. (c) 39.364 and 82.528.

Step by step solution

01

Determine the Degrees of Freedom for Part (a)

The degrees of freedom (df) is calculated as the sample size minus one. Here, the population sample has 16 degrees of freedom.
02

Find the Critical Value for Right-Tailed Test in Part (a)

Using the chi-square distribution table, find the critical value for a right-tailed test with 16 degrees of freedom at the \(\alpha=0.01\) significance level. This value is \(\chi^2_{0.01,16} = 32.000\).
03

Determine the Degrees of Freedom for Part (b)

For part (b), the sample size \(n\) is 14, so the degrees of freedom \(df=n-1=13\).
04

Find the Critical Value for Left-Tailed Test in Part (b)

Using the chi-square distribution table, find the critical value for a left-tailed test with 13 degrees of freedom at \(\alpha=0.01\) significance level. This value is \(\chi^2_{0.99,13} = 3.012\).
05

Determine the Degrees of Freedom for Part (c)

For part (c), the sample size \(n\) is 61, so the degrees of freedom \(df=n-1=60\).
06

Find the Upper and Lower Critical Values for Two-Tailed Test in Part (c)

For a two-tailed test with \(\alpha=0.05\): the upper critical value uses \(\alpha/2=0.025\), so the value is \(\chi^2_{0.025,60} = 82.528\). the lower critical value uses \(\alpha/2=0.025\), so the value is \(\chi^2_{0.975,60} = 39.364\).

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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 are essential in statistical testing. It's the number of values that are free to vary in a calculation without violating any given restrictions. The degrees of freedom (df) for a sample is typically the sample size minus one:
  • For example, if you have a sample size () of 16, then df = 16 - 1 = 15.
  • In another example with a sample size () of 14, df = 14 - 1 = 13.
Knowing the degrees of freedom is crucial because it helps us find the appropriate critical values for our tests.
Right-Tailed Test
A right-tailed test is used when we are interested in determining if a sample mean is significantly greater than the population mean. This means that we look at the upper side of the distribution. For a right-tailed chi-square test, you use a specific point of the chi-square distribution corresponding to the significance level (alpha).
  • For instance, with 16 degrees of freedom and alpha=0.01, the critical value from the chi-square table is 32.000.
  • This means that if our test statistic is greater than 32.000, we reject the null hypothesis.
Left-Tailed Test
A left-tailed test is the opposite of a right-tailed test. Here, we want to check if a sample mean is significantly less than the population mean. We examine the lower side of the distribution. For a left-tailed chi-square test, you use specific critical values from the chi-square distribution table at a given significance level (alpha).
  • With 13 degrees of freedom and alpha=0.01, the critical chi-square value is 3.012.
  • This implies that if our test statistic is less than 3.012, we reject the null hypothesis.
Two-Tailed Test
In a two-tailed test, we are checking for any significant difference, whether it is greater or less than the population mean. This requires us to look at both upper and lower extremes of the distribution. The significance level (alpha) is typically halved.
  • For example, with 60 degrees of freedom and alpha=0.05, each tail will use alpha/2=0.025.
  • The critical values will be obtained from the chi-square table: the upper critical value is 82.528 and the lower critical value is 39.364.
  • If our test statistic falls outside this range, we reject the null hypothesis.
Critical Values
Critical values define the threshold at which we reject the null hypothesis. They depend on the degrees of freedom and the significance level (alpha). For chi-square tests, you refer to the chi-square distribution table to find these values.
  • For example, at alpha=0.01 with 16 degrees of freedom, the right-tailed critical value is 32.000.
  • For a left-tailed test with 13 degrees of freedom at alpha=0.01, the value is 3.012.
  • In a two-tailed test with 60 degrees of freedom and alpha=0.05, the critical values are 82.528 and 39.364.
These values help determine if our test statistic is extreme enough to reject the null hypothesis.
Significance Level
The significance level (alpha) is the probability of rejecting the null hypothesis when it is actually true. Common significance levels are 0.01, 0.05, and 0.1. The lower the alpha, the more stringent the test.
  • For example, at a significance level of 0.01, we have a 1% chance of incorrectly rejecting the null hypothesis.
  • At alpha=0.05, there is a 5% likelihood of making this mistake.
Choosing an appropriate significance level is crucial, as it affects the critical values and ultimately the conclusions drawn from the test.

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Most popular questions from this chapter

According to the National Sleep Foundation, children between the ages of 6 and 11 years should get 10 hours of sleep each night. In a survey of 56 parents of 6 to 11 year olds, it was found that the mean number of hours the children slept was 8.9 with a standard deviation of 3.2. Does the sample data suggest that 6 to 11 year olds are sleeping less than the required amount of time each night? Use the 0.01 level of significance.

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Simulation The exponential probability distribution can be used to model waiting time in line or the lifetime of electronic components. Its density function is skewed right. Suppose the wait time in a line can be modeled by the exponential distribution with \(\mu=\sigma=5\) minutes. (a) Simulate obtaining 100 simple random samples of size \(n=10\) from the population described. That is, simulate obtaining a simple random sample of 10 individuals waiting in a line where the wait time is expected to be 5 minutes. (b) Test the null hypothesis \(H_{0}: \mu=5\) versus the alternative \(H_{1}: \mu \neq 5\) for each of the 100 simulated simple random samples. (c) If we test this hypothesis at the \(\alpha=0.05\) level of significance, how many of the 100 samples would you expect to result in a Type I error? (d) Count the number of samples that lead to a rejection of the null hypothesis. Is it close to the expected value determined in part (c)? What might account for any discrepancies?

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