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Chapter 8: Problem 28

Briefly explain the similarities and the differences between the standard normal distribution and the \(t\) distribution.

Short Answer

Expert verified
Both the standard normal distribution and the t-distribution are symmetric and bell-shaped. However, the standard normal distribution is defined by its mean of 0 and standard deviation of 1, and its shape does not change. The t-distribution, however, has thicker tails which show a higher probability for extreme values and its shape changes with degrees of freedom.

Step by step solution

01

Defining Standard Normal Distribution

The standard normal distribution is a special case of the normal distribution. It is symmetrical and its mean, median, and mode are equal to 0. The shape of a standard normal distribution is entirely defined by its standard deviation, \(\sigma\), which is equal to 1.
02

Defining t-distribution

The t-distribution, on the other hand, is a type of probability distribution that is symmetric and bell-shaped, like the normal distribution. However, its tails are thicker than those of the normal distribution, meaning it has a greater probability for extreme values. The exact shape of the t-distribution depends on its degrees of freedom, which is related to sample size.
03

Comparing the Two Distributions

There are some similarities between the two distributions. Both are symmetric and bell-shaped and defined by their parameters. But they also have differences. The tails of a t-distribution are thicker, indicating a higher probability for extreme values compared to the normal distribution. Also, the shape of a t-distribution changes with degrees of freedom while the shape of a normal distribution is fixed.

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Most popular questions from this chapter

A computer company that recently developed a new software product wanted to estimate the mean time taken to learn how to use this software by people who are somewhat familiar with computers. A random sample of 12 such persons was selected. The following data give the times (in hours) taken by these persons to learn how to use this software. $$ \begin{array}{llllll} 1.75 & 2.25 & 2.40 & 1.90 & 1.50 & 2.75 \\ 2.15 & 2.25 & 1.80 & 2.20 & 3.25 & 2.60 \end{array} $$ Construct a \(95 \%\) confidence interval for the population mean. Assume that the times taken by all persons who are somewhat familiar with computers to learn how to use this software are approximately normally distributed.

For a population, the value of the standard deviation is \(2.65\). A random sample of 35 observations taken from this population produced the following data. \(\begin{array}{lllllll}42 & 51 & 42 & 31 & 28 & 36 & 49 \\ 29 & 46 & 37 & 32 & 27 & 33 & 41 \\ 47 & 41 & 28 & 46 & 34 & 39 & 48 \\ 26 & 35 & 37 & 38 & 46 & 48 & 39 \\ 29 & 31 & 44 & 41 & 37 & 38 & 46\end{array}\) a. What is the point estimate of \(\mu\) ? b. Make a \(98 \%\) confidence interval for \(\mu\). c. What is the margin of error of estimate for part b?

a. Find the value of \(t\) for the \(t\) distribution with a sample size of 21 and area in the left tail equal to \(.10 .\) b. Find the value of \(t\) for the \(t\) distribution with a sample size of 14 and area in the right tail equal to \(.025 .\) c. Find the value of \(t\) for the \(t\) distribution with 45 degrees of freedom and \(.001\) area in the right tail. d. Find the value of \(t\) for the \(t\) distribution with 37 degrees of freedom and \(.005\) area in the left tail.

a. How large a sample should be selected so that the margin of error of estimate for a \(99 \%\) confidence interval for \(p\) is \(.035\) when the value of the sample proportion obtained from a preliminary sample is \(.29 ?\) b. Find the most conservative sample size that will produce the margin of error for a \(99 \%\) confidence interval for \(p\) equal to \(.035\).

A company that produces detergents wants to estimate the mean amount of detergent in 64 -ounce jugs at a \(99 \%\) confidence level. The company knows that the standard deviation of the amounts of detergent in all such jugs is \(.20\) ounce. How large a sample should the company select so that the estimate is within \(.04\) ounce of the population mean?
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