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Chapter 8: Problem 37

What are the parameters of a normal distribution and a \(t\) distribution? Explain.

Short Answer

Expert verified
The normal distribution is defined by two parameters: the mean (\(\mu\)) which determines the center of the distribution, and the standard deviation (\(\sigma\)) which determines the spread or width of the distribution. The t-distribution has one main parameter: the degrees of freedom (\(v\) or \(df\)), which determines the shape of the distribution.

Step by step solution

01

Define a Normal Distribution

A normal distribution, sometimes referred to as Gaussian distribution, is a type of continuous probability distribution for a real-valued random variable. It is a bell-shaped curve described by its mean \(\mu\) and standard deviation \(\sigma\). The mean determines the center of the distribution, and the standard deviation determines the spread or width of the distribution.
02

Define the Parameters of a Normal Distribution

The normal distribution has two parameters: the mean \(\mu\) and the standard deviation \(\sigma\). The mean \(\mu\) is the expectation of the distribution and describes the location of the center of the graph. The standard deviation \(\sigma\) measures the dispersion to the mean or expectation. The variance \(\sigma^2\) is another parameter often used instead of the standard deviation.
03

Define a T-Distribution

The t-distribution, or Student's t-distribution, is a type of probability distribution that is symmetric and bell-shaped like the normal distribution, but has heavier tails. It is used when the sample size is small or the population standard deviation is unknown.
04

Define the Parameters of a T-Distribution

The t-distribution has one main parameter: the degrees of freedom, usually denoted by \(v\) or \(df\). The degrees of freedom contributes to the shape of the graph. As the degrees of freedom increases, the t-distribution approaches the normal distribution.

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Most popular questions from this chapter

A sample selected from a population gave a sample proportion equal to . 73 . a. Make a \(99 \%\) confidence interval for \(p\) assuming \(n=100\). b. Construct a \(99 \%\) confidence interval for \(p\) assuming \(n=600\) c. Make a \(99 \%\) confidence interval for \(p\) assuming \(n=1500\). d. Does the width of the confidence intervals constructed in parts a through \(\mathrm{c}\) decrease as the sample size increases? If yes, explain why.

The mean time taken to design a house plan by 40 architects was found to be 23 hours with a standard deviation of \(3.75\) hours. a. Construct a \(98 \%\) confidence interval for the population mean \(\mu\). b. Suppose the confidence interval obtained in part a is too wide. How can the width of this interval be reduced? Describe all possible alternatives. Which alternative is the best and why?

A department store manager wants to estimate at a \(90 \%\) confidence level the mean amount spent by all customers at this store. The manager knows that the standard deviation of amounts spent by all customers at this store is \(\$ 31\). What sample size should he choose so that the estimate is within \(\$ 3\) of the population mean?

A researcher wanted to know the percentage of judges who are in favor of the death penalty. He took a random sample of 15 judges and asked them whether or not they favor the death penalty. The responses of these iudges are given here. \(\begin{array}{lllllll}\text { Yes } & \text { No } & \text { Yes } & \text { Yes } & \text { No } & \text { No } & \text { No } & \text { Yes } \\ \text { Yes } & \text { No } & \text { Yes } & \text { Yes } & \text { Yes } & \text { No } & \text { Yes }\end{array}\) a. What is the point estimate of the population proportion? b. Make a \(95 \%\) confidence interval for the percentage of all judges who are in favor of the death penalty.

The following data give the speeds (in miles per hour), as measured by radar, of 10 cars traveling on Interstate \(\mathrm{I}-15\). \(\begin{array}{llllllllll}76 & 72 & 80 & 68 & 76 & 74 & 71 & 78 & 82 & 65\end{array}\) Assuming that the speeds of all cars traveling on this highway have a normal distribution, construct a \(90 \%\) confidence interval for the mean speed of all cars traveling on this highway.
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