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Chapter 3: Problem 55

Briefly explain the empirical rule. To what kind of distribution is it applied?

Short Answer

Expert verified
The Empirical Rule, also called the 68-95-99.7 rule, states that within a normal distribution nearly all data falls within three standard deviations of the mean. Specifically, about 68% of data falls within one standard deviation, 95% within two standard deviations, and 99.7% within three standard deviations. This rule is applied to Normal Distributions, which are symmetrical and have a bell-shaped curve.

Step by step solution

01

Definition of Empirical Rule

The Empirical Rule, also known as the 68-95-99.7 rule, is a rule which states that for a normal distribution, almost all data falls within three standard deviations (蟽) of the mean (渭). More precisely, about 68% of the data falls within one standard deviation, 95% falls within two standard deviations, and 99.7% falls within three standard deviations.
02

Application of Empirical Rule

The Empirical Rule is applied to a specific type of distribution called the Normal Distribution. A normal distribution is a type of continuous probability distribution for a real-valued random variable. It is symmetric and its mean, median and mode are equal.
03

Properties of Normal Distribution

In a Normal Distribution, data is distributed in a way that the mean, median, and mode are all the same and fall in the center of the distribution. The curve of the distribution is bell-shaped and symmetrical about the mean. Importantly, approximately 68% of the data falls within one standard deviation, about 95% within two standard deviations, and about 99.7% within three standard deviations from the mean, which is essentially the empirical rule.

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Most popular questions from this chapter

The following data give the time (in minutes) that each of 20 students selected from a university waited in line at their bookstore to pay for their textbooks in the beginning of the Fall 2015 semester. \(\begin{array}{rrrrrrrrrr}15 & 8 & 23 & 21 & 5 & 17 & 31 & 22 & 34 & 6 \\ 5 & 10 & 14 & 17 & 16 & 25 & 30 & 3 & 31 & 19\end{array}\) Prepare a box-and-whisker plot. Comment on the skewness of these data.

Prices of cars have a distribution that is skewed to the right with outliers in the right tail. Which of the measures of center is the best to summarize this data set? Explain.

The following data give the amounts (in dollars) of electric bills for November 2015 for 12 randomly selected households selected from a small town. \(\begin{array}{llllllllllll}205 & 265 & 176 & 314 & 243 & 192 & 297 & 357 & 238 & 281 & 342 & 259\end{array}\) Calculate the mean and median for these data. Do these data have a mode? Explain.

The following data give the total food expenditures (in dollars) for the past one month for a sample of 20 families. \(\begin{array}{rrrrrrrrrr}1125 & 530 & 1234 & 595 & 427 & 872 & 1480 & 699 & 1274 & 1187 \\ 933 & 1127 & 716 & 1065 & 934 & 930 & 1046 & 1199 & 1353 & 441\end{array}\) a. Calculate the mean and median for these data. b. Calculate the \(20 \%\) trimmed mean for these data.

Briefly explain the meaning of an outlier. Is the mean or the median a better measure of center for a data set that contains outliers? Illustrate with the help of an example.
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