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Chapter 8: Problem 16

Let the test statistic \(T\) have a \(t\) distribution when \(H_{0}\) is true. Give the significance level for each of the following situations: a. \(H_{\mathrm{a}}: \mu>\mu_{0}, \mathrm{df}=15\), rejection region \(t \geq 3.733\) b. \(H_{\mathrm{a}}: \mu<\mu_{0}, n=24\), rejection region \(t \leq-2.500\) c. \(H_{\mathrm{a}}: \mu \neq \mu_{0}, n=31\), rejection region \(t \geq 1.697\) or \(t \leq-1.697\)

Short Answer

Expert verified
(a) 0.001, (b) 0.01, (c) 0.1

Step by step solution

01

Understanding the Problem

We need to find the significance level for different hypothesis testing situations. The significance level is the probability of rejecting the null hypothesis when it is true and is denoted by \( \alpha \). In each scenario, observe the degrees of freedom (\( \text{df} \)) and use the given rejection region to find \( \alpha \).
02

Calculate Significance Level for Situation (a)

For situation (a), the alternative hypothesis is \( H_a: \mu > \mu_0 \). This is a one-tailed test with \( \text{df} = 15 \). The rejection region is \( t \geq 3.733 \). Use the t-distribution table to find the probability that a \( t \) value with 15 degrees of freedom is greater than 3.733. This probability is the significance level \( \alpha \). After consulting the table, \( \alpha = 0.001 \).
03

Calculate Significance Level for Situation (b)

For situation (b), the alternative hypothesis is \( H_a: \mu < \mu_0 \). The rejection region is \( t \leq -2.500 \) with \( n=24 \), meaning \( \text{df} = 23 \) (since \( \text{df} = n-1 \)). Look up a t-distribution table for \( \text{df} = 23 \) to find the probability of \( t \leq -2.500 \). This probability is the significance level \( \alpha \). From the table, \( \alpha = 0.01 \).
04

Calculate Significance Level for Situation (c)

For situation (c), the alternative hypothesis is \( H_a: \mu eq \mu_0 \), which is a two-tailed test where the rejection region is \( t \geq 1.697 \) or \( t \leq -1.697 \). With \( n=31 \), \( \text{df} = 30 \). Find the probability for both tails using the t-distribution table for \( \text{df} = 30 \) and \( |t| \geq 1.697 \). The sum of these probabilities gives the significance level \( \alpha \). From the table, \( \alpha = 0.1 \).

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Key Concepts

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

t-distribution
A t-distribution is a type of probability distribution that is theoretical in nature and essential for performing hypothesis testing, especially when working with smaller sample sizes or when the population standard deviation is unknown.
This distribution is symmetric and bell-shaped, similar to the normal distribution, but has heavier tails. Heavier tails mean there is a greater probability of values far from the mean. This is particularly useful in hypothesis testing because it provides a more realistic model for data that is less predictable.
The t-distribution depends on degrees of freedom (df), which affects its shape:
  • Lower degrees of freedom result in a distribution with more pronounced tails and a flatter peak.
  • As the degrees of freedom increase, the t-distribution approaches the shape of a standard normal distribution.
Knowing the t-distribution is crucial when determining the significance level and critical t-values from statistical tables, which in turn helps decide whether to reject the null hypothesis.
significance level
The significance level, often denoted by \( \alpha \), is a fundamental concept in hypothesis testing. It represents the probability of rejecting the null hypothesis when it is actually true, which is a type I error.
The choice of significance level can affect the results of a hypothesis test. Common choice values for \( \alpha \) include 0.05, 0.01, and 0.10. A lower significance level means that you require more substantial evidence against the null hypothesis before you decide to reject it.
In practical terms, the significance level defines the rejection region, which is the range of values for which we invalidate the null hypothesis:
  • For a one-tailed test, the rejection region is on one side of the distribution (either large or small values).
  • For a two-tailed test, the rejection region is split between the two tails, accommodating both significantly large and small values.
Understanding how to calculate and interpret the significance level is critical for making informed decisions based on statistical evidence.
degrees of freedom
Degrees of freedom (df) are a key component in statistical calculations and especially crucial when referencing the t-distribution table. They represent the number of independent values or quantities that can vary within the statistical calculation without breaking any constraints.
In the context of hypothesis testing with a t-distribution, degrees of freedom typically depend on the sample size \( n \):
  • For simple t-tests, the formula for degrees of freedom is usually \( df = n - 1 \).
  • More complex models might use different calculations for df, considering factors like the number of groups or variables.
The degrees of freedom influence the shape of the t-distribution curve. When interpreting results from tables, a correct understanding of df ensures precise access to critical t-values, which in turn determines the rejection criteria for hypothesis testing.

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Most popular questions from this chapter

Let \(X_{1}, \ldots, X_{n}\) denote a random sample from a normal population distribution with a known value of \(\sigma\). a. For testing the hypotheses \(H_{0}: \mu=\mu_{0}\) versus \(H_{\mathrm{a}}: \mu>\mu_{0}\) (where \(\mu_{0}\) is a fixed number), show that the test with test statistic \(\bar{X}\) and rejection region \(\bar{x} \geq \mu_{0}+2.33 \sigma / \sqrt{n}\) has significance level \(.01\). b. Suppose the procedure of part (a) is used to test \(H_{0}: \mu \leq \mu_{0}\) versus \(H_{\mathrm{a}}: \mu>\mu_{0}\). If \(\mu_{0}=100, n=25\), and \(\sigma=5\), what is the probability of committing a type I error when \(\mu=99\) ? When \(\mu=98\) ? In general, what can be said about the probability of a type I error when the actual value of \(\mu\) is less than \(\mu_{0}\) ? Verify your assertion.

A new design for the braking system on a certain type of car has been proposed. For the current system, the true average braking distance at \(40 \mathrm{mph}\) under specified conditions is known to be \(120 \mathrm{ft}\). It is proposed that the new design be implemented only if sample data strongly indicates a reduction in true average braking distance for the new design. a. Define the parameter of interest and state the relevant hypotheses. b. Suppose braking distance for the new system is normally distributed with \(\sigma=10\). Let \(\bar{X}\) denote the sample average braking distance for a random sample of 36 observations. Which of the following three rejection regions is appropriate: \(R_{1}=\\{\bar{x}: \bar{x} \geq 124.80\\}, \quad R_{2}=\\{\bar{x}: \bar{x} \leq 115.20\\}\), \(R_{3}=\\{\bar{x}\) : either \(\bar{x} \geq 125.13\) or \(\bar{x} \leq 114.87\\}\) ? c. What is the significance level for the appropriate region of part (b)? How would you change the region to obtain a test with \(\alpha=.001\) ? d. What is the probability that the new design is not implemented when its true average braking distance is actually \(115 \mathrm{ft}\) and the appropriate region from part (b) is used? e. Let \(Z=(\bar{X}-120) /(\sigma / \sqrt{n})\). What is the significance level for the rejection region \(\\{z: z \leq-2.33\\}\) ? For the region \(\\{z: z \leq-2.88\\}\) ?

Have you ever been frustrated because you could not get a container of some sort to release the last bit of its contents? The article "Shake, Rattle, and Squeeze: How Much Is Left in That Container?" (Consumer Reports, May 2009: 8) reported on an investigation of this issue for various consumer products. Suppose five \(6.0 \mathrm{oz}\) tubes of toothpaste of a particular brand are randomly selected and squeezed until no more toothpaste will come out. Then each tube is cut open and the amount remaining is weighed, resulting in the following data (consistent with what the cited article reported): \(.53, .65, .46, .50, .37\). Does it appear that the true average amount left is less than \(10 \%\) of the advertised net contents? a. Check the validity of any assumptions necessary for testing the appropriate hypotheses. b. Carry out a test of the appropriate hypotheses using a significance level of \(.05\). Would your conclusion change if a significance level of .01 had been used? c. Describe in context type I and II errors, and say which error might have been made in reaching a conclusion.

Exercise 36 in Chapter 1 gave \(n=26\) observations on escape time \((\mathrm{sec})\) for oil workers in a simulated exercise, from which the sample mean and sample standard deviation are \(370.69\) and \(24.36\), respectively. Suppose the investigators had believed a priori that true average escape time would be at most \(6 \mathrm{~min}\). Does the data contradict this prior belief? Assuming normality, test the appropriate hypotheses using a significance level of .05.

An article in the Nov. 11,2005 , issue of the San Luis Obispo Tribune reported that researchers making random purchases at California Wal-Mart stores found scanners coming up with the wrong price \(8.3 \%\) of the time. Suppose this was based on 200 purchases. The National Institute for Standards and Technology says that in the long run at most two out of every 100 items should have incorrectly scanned prices. a. Develop a test procedure with a significance level of (approximately) .05, and then carry out the test to decide whether the NIST benchmark is not satisfied. b. For the test procedure you employed in (a), what is the probability of deciding that the NIST benchmark has been satisfied when in fact the mistake rate is \(5 \%\) ?
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