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Chapter 9: Problem 7

(a) Find the \(t\) -value such that the area in the right tail is 0.10 with 25 degrees of freedom. (b) Find the \(t\) -value such that the area in the right tail is 0.05 with 30 degrees of freedom. (c) Find the \(t\) -value such that the area left of the \(t\) -value is 0.01 with 18 degrees of freedom. [Hint: Use symmetry. (d) Find the critical \(t\) -value that corresponds to \(90 \%\) confidence. Assume 20 degrees of freedom.

Short Answer

Expert verified
a) 1.316 b) 1.697 c) -2.552 d) 1.725

Step by step solution

01

- Understand the Problem

The problem requires the finding of the critical t-values for different confidence levels and degrees of freedom. Knowing which t-distribution table or software to use for these calculations is vital.
02

- Find the t-value for part (a)

Use the t-distribution table or a calculator. For the area in the right tail being 0.10 with 25 degrees of freedom, look up the t-value corresponding to 0.90 in the t-distribution table (since right tail of 0.10 implies a cumulative area of 0.90 to the left). The t-value is approximately 1.316.
03

- Find the t-value for part (b)

For the area in the right tail being 0.05 with 30 degrees of freedom, find the t-value corresponding to 0.95 in the t-distribution table. This t-value is approximately 1.697.
04

- Find the t-value for part (c)

For the area left of the t-value being 0.01 with 18 degrees of freedom, use the symmetry property of the t-distribution. Find the t-value corresponding to 0.99 in the right tail, which is the same as -t for the 0.01 left tail. This t-value is approximately -2.552.
05

- Find the critical t-value for part (d)

For a 90% confidence interval with 20 degrees of freedom, the critical t-value corresponds to the t-value with 5% in each tail. Hence, lookup 0.95 in the t-distribution table for 20 degrees of freedom. The critical t-value is approximately 1.725.

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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 (df) are a concept used in various statistical analyses. They represent the number of values in a calculation that are free to vary. When estimating a parameter, like the mean of a population, degrees of freedom are the number of independent pieces of information available. For example, if you are using a sample of size 25 to estimate the mean, you have 24 degrees of freedom (since one piece of information is used up by the sample mean itself). The degrees of freedom impact the shape and width of the t-distribution. The more degrees of freedom you have, the closer the t-distribution gets to the normal distribution.
Critical t-value
The critical t-value is a threshold in the t-distribution that defines the boundary for a given level of confidence or significance. It's the value beyond which the probability in the tail is equal to a specified significance level, such as 0.05 or 0.01. You can find the critical t-value using t-distribution tables or software. When you look up a critical t-value, you need to supply the degrees of freedom and the desired tail area. For instance, in part (a) of the problem, with 25 degrees of freedom and a right tail area of 0.10, the critical t-value is about 1.316. This t-value tells us that for this many degrees of freedom, only 10% of the data falls beyond this value.
Confidence Interval
A confidence interval (CI) is a range of values derived from sample data that is likely to contain the true population parameter, such as the mean, a certain percentage of the time. The width of this interval depends on the sample size and the variability in the data. For example, a 90% confidence interval means that if we were to take 100 different samples and compute a CI for each, we expect approximately 90 of those intervals to include the population parameter. In part (d) of the problem, we use the critical t-value to find the 90% confidence interval for a given dataset with 20 degrees of freedom. We look up the t-value for 0.95 (since 5% in each tail), which is 1.725.
Right Tail Area
The right tail area of a t-distribution represents the probability of observing a value greater than a given critical t-value. This area is crucial when you're conducting hypothesis tests. For example, when the problem asks for the t-value such that the area in the right tail is 0.10 with 25 degrees of freedom, it means we need to find the t-value where 10% of the distribution lies to the right. This corresponds to a cumulative area (from the left up to that point) of 0.90. So, for part (a), we find the t-value that leaves 10% in the right tail, which is about 1.316.
Symmetry in Distribution
The t-distribution is symmetrical around its mean, which is zero. This symmetry property is helpful when finding t-values for both left and right tails. When given an area in one tail, you can use symmetry to understand the other tail. For instance, if you know the area in the right tail is 0.05, the corresponding area in the left tail is also 0.05 (since the total area under the curve is 1). In part (c) of the problem, it asks for the t-value such that the area to the left is 0.01 with 18 degrees of freedom. You look up the t-value for 0.99 in the right tail, which by symmetry means the 鈭抰 value for the 0.01 left tail. This t-value is about 鈭�2.552.

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Most popular questions from this chapter

A simple random sample of size \(n\) is drawn from a population that is normally distributed with population standard deviation, \(\sigma,\) known to be \(13 .\) The sample mean, \(\bar{x},\) is found to be 108. (a) Compute the \(96 \%\) confidence interval about \(\mu\) if the sample size, \(n,\) is 25 (b) Compute the \(96 \%\) confidence interval about \(\mu\) if the sample size, \(n\), is \(10 .\) How does decreasing the sample size affect the margin of error, \(E ?\) (c) Compute the \(88 \%\) confidence interval about \(\mu\) if the sample size, \(n,\) is \(25 .\) Compare the results to those obtained in part (a). How does decreasing the level of confidence affect the size of the margin of error, \(E ?\) (d) Could we have computed the confidence intervals in parts (a)-(c) if the population had not been normally distributed? Why? (e) Suppose an analysis of the sample data revealed three outliers greater than the mean. How would this affect the confidence interval?

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A simple random sample of size \(n\) is drawn from a population that is normally distributed. The sample mean, \(\bar{x}\), is found to be \(50,\) and the sample standard deviation, \(s,\) is found to be \(8 .\) (a) Construct a \(98 \%\) confidence interval about \(\mu\) if the sample size, \(n,\) is \(20 .\) (b) Construct a \(98 \%\) confidence interval about \(\mu\) if the sample size, \(n,\) is \(15 .\) How does decreasing the sample size affect the margin of error, \(E ?\) (c) Construct a \(95 \%\) confidence interval about \(\mu\) if the sample size, \(n,\) is \(20 .\) Compare the results to those obtained in part (a). How does decreasing the level of confidence affect the margin of error, \(E ?\) (d) Could we have computed the confidence intervals in parts (a)-(c) if the population had not been normally distributed? Why?

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Construct the appropriate confidence interval. In a random sample of 40 felons convicted of aggravated assault, it was determined that the mean length of sentencing was 54 months, with a standard deviation of 8 months. Construct and interpret a \(95 \%\) confidence interval for the mean length of sentence for an aggravated assault conviction. (Source: Based on data obtained from the U.S. Department of Justice.)
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