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Chapter 9: Problem 8

(a) Find the \(t\) -value such that the area in the right tail is 0.02 with 19 degrees of freedom. (b) Find the \(t\) -value such that the area in the right tail is 0.10 with 32 degrees of freedom. (c) Find the \(t\) -value such that the area left of the \(t\) -value is 0.05 with 6 degrees of freedom. [Hint: Use symmetry.] (d) Find the critical \(t\) -value that corresponds to \(95 \%\) confidence. Assume 16 degrees of freedom.

Short Answer

Expert verified
a) 2.539b) 1.310c) -1.943d) 2.120

Step by step solution

01

Identify the given parameters

For each part of the problem, write down the given area and the degrees of freedom. For example, in part (a), the area in the right tail is 0.02 and the degrees of freedom are 19.
02

Use the t-distribution table or calculator

For part (a), look up the t-value for an area of 0.02 in the right tail with 19 degrees of freedom using a t-distribution table or a calculator. The same method applies to parts (b), (c), and (d).
03

Calculate the t-value for each part

a) For an area of 0.02 in the right tail with 19 degrees of freedom, the t-value is approximately 2.539.b) For an area of 0.10 in the right tail with 32 degrees of freedom, the t-value is approximately 1.310.c) For an area of 0.05 left of the t-value with 6 degrees of freedom, use the hint about symmetry. If the left tail area is 0.05, the right tail area is 0.95. The t-value is approximately -1.943.d) For a 95% confidence interval with 16 degrees of freedom, the critical t-value is approximately 2.120. This means the region in each tail is 0.025.

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Key Concepts

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

Right Tail Area
In statistics, the right tail area of a t-distribution refers to the probability that a t-value is greater than a specific value. It is the area under the curve to the right of that t-value. Calculating this area helps determine the corresponding t-value for various degrees of freedom. For example, if the area in the right tail is 0.02 with 19 degrees of freedom, the t-value is approximately 2.539.

To find this t-value:
  • Identify the right tail area and degrees of freedom from the problem.
  • Use a t-distribution table or a calculator.
The t-value indicates the number of standard deviations a data point is from the mean, specific to smaller sample sizes.
Degrees of Freedom
Degrees of freedom (df) refer to the number of independent values in a calculation that are free to vary. In the context of t-distributions, the degrees of freedom are typically calculated as the sample size (n) minus one: df = n - 1.

When calculating t-values:
  • Identify the sample size.
  • Subtract 1 to find the degrees of freedom.
For instance, if a sample has 20 values, the degrees of freedom would be 19. Degrees of freedom impact the shape of the t-distribution, with more degrees of freedom leading to distributions that closely resemble the standard normal distribution.
95% Confidence Interval
A 95% Confidence Interval (CI) indicates the range within which we can be 95% certain that the population mean lies, based on the sample mean and standard error. To find this interval, we use the critical t-value associated with 95% confidence and the appropriate degrees of freedom.

For example:
  • The total area under the curve is 1 (100%).
  • A 95% CI corresponds to 0.95 area in the middle, leaving 0.025 in each tail.
Thus, the critical t-value for 16 degrees of freedom is approximately 卤2.120, as found using tables or calculators. This t-value helps calculate the margin for error in the confidence interval.
Critical t-value
The critical t-value is a threshold in the t-distribution that corresponds to a specific confidence level or tail area. It is used to determine the cut-off points for confidence intervals or hypothesis tests.

Finding the critical t-value involves:
  • Identifying the desired confidence level or tail area.
  • Determining the degrees of freedom.
  • Using a t-distribution table or software to find the corresponding t-value.
For instance, with 32 degrees of freedom and a right tail area of 0.10, the critical t-value is approximately 1.310.
Symmetry in T-Distribution
The t-distribution is symmetric around zero, meaning it has mirror-image left and right tails. This property lets us use the right tail t-values to find left tail t-values by changing the sign. For example:
  • If the left tail area is 0.05 (areas less than the t-value), the corresponding right tail area is 0.95.
  • Use the symmetry by finding the positive t-value for the right tail and then negating it for the left tail.
Thus, for 6 degrees of freedom, an area of 0.05 in the left tail corresponds to a t-value of -1.943 by symmetry.

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Most popular questions from this chapter

According to Nielsen/ NetRatings, the mean amount of time spent on MySpace.com per user per month in July 2014 was 171.0 minutes. A \(95 \%\) confidence interval for the mean amount of time spent on MySpace.com monthly has a lower bound of 151.4 minutes and an upper bound of 190.6 minutes. What does this mean?

The Sullivan Statistics Survey II asks, "What percent of one's income should an individual pay in federal income taxes?" Go to www.pearsonhighered.com/sullivanstats to obtain the data file SullivanStatsSurveyII using the file format of your choice for the version of the text you are using. The data is in the column "Tax Rate" (a) Draw a relative frequency histogram of the variable "Tax Rate" using a lower class limit of the first class of 0 and a class width of \(5 .\) Comment on the shape of the distribution. (b) Draw a boxplot of the variable "Tax Rate." Are there any outliers? (c) Explain why a large sample is necessary to construct a confidence interval for the mean tax rate. (d) Treat the respondents of this survey as a simple random sample of U.S. residents. Use statistical software to construct and interpret a \(90 \%\) confidence interval for the mean tax rate U.S. residents feel an individual should pay in federal income taxes.

Construct the appropriate confidence interval. A simple random sample of size \(n=25\) is drawn from a population that is normally distributed. The sample variance is found to be \(s^{2}=3.97\). Construct a \(95 \%\) confidence interval for the population standard deviation.

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In a Gallup poll, \(64 \%\) of the people polled answered yes to the following question: "Are you in favor of the death penalty for a person convicted of murder?" The margin of error in the poll was \(3 \%,\) and the estimate was made with \(95 \%\) confidence. At least how many people were surveyed?
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