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Chapter 8: Q. R8.9 (page 548)

t time When constructing confidence intervals for a population mean, we almost always use critical values from a t distribution rather than the standard Normal distribution.

a. When is it necessary to use a t critical value rather than a z critical value when constructing a confidence interval for a population means?

b. For a particular level of confidence, explain what happens to the t critical values as Page Number: 548 the degrees of freedom increase.

Short Answer

Expert verified

Part a) When the sample standard deviation is known and the sample size is less than30, the t-distribution is used to calculate the critical value.

Part b) It could be said that as the degree of freedom increases, the tcritical value approaches the zcritical value.

Step by step solution

01

Part a) Step 1: The objective is to explain the condition required to construct the confidence interval using the t critical value rather than z

When the standard deviation is known, the Z-distribution is used to compute the confidence interval for the population mean.

When the sample standard deviation is known and the sample size is less than30, thet-distribution is used to calculate the critical value.

02

Part b) Step 2: The objective is to explain the effect the increase of the degree of freedom on tcritical value for a particular confidence level.

One of the t-properties distributions is that as the degree of freedom increases, the t-distribution approaches the normal distribution.

Therefore, It could be said that as the degree of freedom increases, the tcritical value approaches the zcritical value.

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Most popular questions from this chapter

Check them all Determine if the conditions are met for constructing a confidence interval for the population mean in each of the following settings.

a. We want to estimate the average age at which U.S. presidents have died. So we obtain a list of all U.S. presidents who have died and their ages at death.

b. Do teens text more than they call? To find out, an AP Statistics class at a large high school collected data on the number of text messages and calls sent or received by each of 25randomly selected students. The boxplot displays the difference (Texts − Calls) for each student.

How confident? The figure shows the result of taking 25SRSs from a Normal population and constructing a confidence interval for the population mean using each sample. Which confidence level—role="math" localid="1654197644211" 80%,90%,95%,or99%—do you think was used? Explain your reasoning.

Explaining confidence The admissions director for a university found that (107.8,116.2)is a 95%confidence interval for the mean IQ score of all freshmen. Discuss whether each of the following explanations is correct, based on that information.

a. There is a 95%probability that the interval from role="math" localid="1654200953396" 107.8to116.2contains μ.

b. There is a 95%chance that the interval(107.8,116.2)contains x¯.

c. This interval was constructed using a method that produces intervals that capture the true mean in 95%of all possible samples.

d. If we take many samples, about 95%of them will contain the interval (107.8,116.2).

e. The probability that the interval (107.8,116.2)captures μ is either 0 or 1, but we don’t know which.

Gambling and the NCAA Gambling is an issue of great concern to those involved in college athletics. Because of this concern, the National Collegiate Athletic Association (NCAA) surveyed randomly selected student athletes concerning their gambling-related behaviors. Of the 5594Division Imale athletes who responded to the survey, 3547reported participation in some gambling behavior. This includes playing cards, betting on games of skill, buying lottery tickets, betting on sports, and similar activities. A report of this study cited a 1%margin of error.

a. The confidence level was not stated in the report. Use what you have learned to

estimate the confidence level, assuming that the NCAA took an SRS.

b. The study was designed to protect the anonymity of the student athletes who responded. As a result, it was not possible to calculate the number of students who were asked to respond but did not. How does this fact affect the way that you interpret the results?

You have an SRS of 23observations from a large population. The distribution of sample values is roughly symmetric with no outliers. What critical value would you use to obtain a 98%confidence interval for the mean of the population?

a. 2.177

b. 2.183

c. 2.326

d. 2.500

e. 2.508
See all solutions

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