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Chapter 8: Problem 36

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

Short Answer

Expert verified
The standard normal and \(t\) distribution are both symmetrical and bell-shaped. The key differences are that the standard normal distribution has thinner tails and is used when population parameters are known, whereas the \(t\)-distribution has thicker tails and is used when these parameters are unknown and estimated from the sample data.

Step by step solution

01

Describing Standard Normal Distribution

Standard normal distribution, also known as Z-distribution, is a special case of normal distribution where the mean (\(\mu\)) is 0 and standard deviation (\(\sigma\)) is 1. The total area under its probability density function is 1. It is symmetrical around 0 and follows the empirical rule, which states that almost all observed data will fall within three standard deviations of the mean.
02

Describing \(t\) Distribution

The \(t\) distribution, also known as Student鈥檚 t-distribution, is a type of probability distribution that is symmetric and bell-shaped, much like the standard normal distribution. However, it has heavier tails, which means it is more prone to having values that fall far from its mean. The shape of the \(t\) distribution depends on its degrees of freedom. As the degrees of freedom increase, the \(t\) distribution approaches a standard normal distribution.
03

Comparing Both Distributions

Both distributions are symmetrical and bell-shaped. However, the key difference between them is the shape of their tails. The standard normal distribution has a more peaked distribution around the mean and thinner tails, implying less probability for extreme values. On the other hand, the \(t\)-distribution has flatter peak and fatter tails, which allows for greater possibility for extreme values. Also, the standard normal distribution is used when population parameters (\(\mu\) and \(\sigma\)) are known, whereas the \(t\) distribution is typically used when these are not known and are estimated from sample data.

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Most popular questions from this chapter

For each of the following, find the area in the appropriate tail of the \(t\) distribution. a. \(t=2.467\) and \(d f=28\) b. \(t=-1.672\) and \(d f=58\) c. \(t=-2.670\) and \(n=55\) d. \(t=2.383\) and \(n=23\)

a. Find the value of \(t\) from the \(t\) distribution table for a sample size of 22 and a confidence level of \(95 \%\) b. Find the value of \(t\) from the \(t\) distribution table for 60 degrees of freedom and a \(90 \%\) confidence level. c. Find the value of \(t\) from the \(t\) distribution table for a sample size of 24 and a confidence level of \(99 \%\)

It is said that happy and healthy workers are efficient and productive. A company that manufactures exercising machines wanted to know the percentage of large companies that provide on-site health club facilities. A sample of 240 such companies showed that 96 of them provide such facilities on site. a. What is the point estimate of the percentage of all such companies that provide such facilities on site? b. Construct a \(97 \%\) confidence interval for the percentage of all such companies that provide such facilities on site. What is the margin of error for this estimate?

A sample of 18 observations taken from a normally distributed population produced the following data. \(\begin{array}{lllllllll}28.4 & 27.3 & 25.5 & 25.5 & 31.1 & 23.0 & 26.3 & 24.6 & 28.4\end{array}\) \(\begin{array}{llllllll}37.2 & 23.9 & 28.7 & 27.9 & 25.1 & 27.2 & 25.3\end{array}\) \(\begin{array}{ll}22.6 & 22.7\end{array}\) a. What is the point estimate of \(\mu\) ? b. Make a \(99 \%\) confidence interval for \(\mu .\) c. What is the margin of error of estimate for \(\mu\) in part b?

When calculating a confidence interval for the population mean \(\mu\) with a known population standard deviation \(\sigma\), describe the effects of the following two changes on the confidence interval: (1) doubling the sample size, (2) quadrupling (multiplying by 4) the sample size. Give two reasons why this relationship does not hold true if you are calculating a confidence interval for the population mean \(\mu\) with an unknown population standard deviation.
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