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Chapter 3: Problem 84

The prices of all college textbooks follow a bell-shaped distribution with a mean of \(\$ 180\) and a standard deviation of \(\$ 30\). A. Using the empirical rule, find the percentage of all college textbooks with their prices between 1\. \(\$ 150\) and \(\$ 210\) ii. \(\$ 120\) and \(\$ 240\) *b. Using the empirical rule, find the interval that contains the prices of \(99.7 \%\) of college textbooks.

Short Answer

Expert verified
a) Approximately 68% of all textbook prices fall between \$150 and \$210, while approximately 95% of textbook prices fall between \$120 and \$240. b) The price range that contains 99.7% of all college textbooks is between \$90 and \$270.

Step by step solution

01

Applying the Empirical Rule to find percentage

a) The empirical rule states that: \n - approximately 68% of the data falls within 1 standard deviation of the mean. \n - approximately 95% falls within 2 standard deviations. \n - approximately 99.7% falls within 3 standard deviations. \n The price range from \$150 to \$210 is within 1 standard deviation of the mean (mean 卤 \$30). Therefore, approximately 68% of all college textbook prices should fall within this range. \n The second range, from \$120 to \$240, is within 2 standard deviations of the mean (mean 卤 (2 * \$30)). Therefore, approximately 95% of the textbook prices should fall within this range.
02

Applying the Empirical Rule to find interval

b) The empirical rule says that for a normal distribution, almost all (or about 99.7%) of the data will fall within 3 standard deviations of the mean. \n Here, the mean is \$180 and the standard deviation is \$30. To find the interval for 99.7% of the dataset, we calculate: \n - Lower limit = Mean - 3 * Standard deviation = \$180 - 3*\$30 = \$90 \n - Upper limit = Mean + 3 * Standard deviation = \$180 + 3*\$30 = \$270 \n Thus, the price range that contains 99.7% of all college textbooks should be between \$90 and \$270.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Bell-shaped distribution
Imagine plotting the prices of all college textbooks on a graph. If done correctly, you'll notice that the graph has a distinct, symmetrical shape resembling a bell. That's why it's called a bell-shaped distribution. This type of distribution is typical for many naturally occurring data sets.

The highest frequency of data points, or prices in this case, will be around the mean price of the textbooks, and this is where the bell is the tallest. As the prices move away from the mean, either higher or lower, the frequency gets smaller, creating the bell's sloping sides. This characteristic helps in using statistical tools, like the Empirical Rule, to make sense of data.
Standard deviation
Standard deviation is a measure of how much variation exists from the mean in a set of data. In our example, the standard deviation of the textbook prices is $30. This means, when prices deviate from the average price of $180, they usually do so by around $30.

Think of standard deviation as a "spread index". A small standard deviation means that data points are generally close to the mean, while a large standard deviation indicates that they are more spread out. When you know the standard deviation, you have a better picture of the data's layout and how prices vary across textbooks.
Percentages within standard deviations
The Empirical Rule provides a way to predict how data points are spread in a bell-shaped distribution. According to this rule:

  • Approximately 68% of data points lie within 1 standard deviation of the mean.
  • Approximately 95% of data points lie within 2 standard deviations of the mean.
  • Approximately 99.7% lie within 3 standard deviations.

For the textbook prices, moving 1 standard deviation ($30) from the mean ($180) covers $150 to $210, where about 68% of textbook prices fall. Extending to 2 standard deviations (a range of $120 to $240) captures 95% of prices. This predictive power makes understanding data much easier and is crucial for anyone dealing with statistics.
Statistical intervals
When we talk about statistical intervals, we're referring to specific ranges predicted by statistical rules like the Empirical Rule. These intervals tell us where most of the data points in a distribution are likely to be found.

In the context of textbook prices, knowing that 99.7% of them lie between $90 and $270 offers a comprehensive view. This interval (mean 卤 three standard deviations) covers almost all the data, making it easier to handle outliers or extremes in the dataset. Understanding these intervals helps in planning, decision-making, and predicting scenarios efficiently.

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Most popular questions from this chapter

A local golf club has men's and women's summer leagues. The following data give the scores for 芒 round of 18 holes of golf for 17 men and 15 women randomly selected from their respective leagues. \begin{tabular}{l|rrrrrrrrr} \hline Men & 87 & 68 & 92 & 79 & 83 & 67 & 71 & 92 & 112 \\ & 75 & 77 & 102 & 79 & 78 & 85 & 75 & 72 & \\ \hline Women & 101 & 100 & 87 & 95 & 98 & 81 & 117 & 107 & 103 \\ & 97 & 90 & 100 & 99 & 94 & 94 & & & \\ \hline \end{tabular} a. Make a box-and-whisker plot for each of the data sets and use them to discuss the similarities and differences between the scores of the men and women golfers. b. Compute the various descriptive measures you have learned for each sample. How do they compare?

The following data give the numbers of text messages sent by a high school student on 40 randomly selected days during 2012: \(\begin{array}{llllllllll}32 & 33 & 33 & 34 & 35 & 36 & 37 & 37 & 37 & 37 \\\ 38 & 39 & 40 & 41 & 41 & 42 & 42 & 42 & 43 & 44 \\ 44 & 45 & 45 & 45 & 47 & 47 & 47 & 47 & 47 & 48 \\ 48 & 49 & 50 & 50 & 51 & 52 & 53 & 54 & 59 & 61\end{array}\) a. Calculate the values of the three quartiles and the interquartile range. Where does the value 49 fall in relation to these quartiles? b. Determine the approximate value of the 91 st percentile. Give a brief interpretation of this percentile. c. For what percentage of the days was the number of text messages sent 40 or higher? Answer by finding the percentile rank of 40 .

The mean life of a certain brand of auto batteries is 44 months with a standard deviation of 3 months. Assume that the lives of all auto batteries of this brand have a bell-shaped distribution. Using the empirical rule, find the percentage of auto batteries of this brand that have a life of if. 41 to 47 months b. 38 to 50 months c. 35 to 53 months

The trimmed mean is calculated by dropping a certain percentage of values from each end of a ranked data set. The trimmed mean is especially useful as a measure of central tendency when a data set contains a few outliers. Suppose the following data give the ages (in years) of 10 employees of a company: \(\begin{array}{llllll}47 & 53 & 38 & 26 & 39 & 49\end{array}\) 19 \(\begin{array}{ccc}67 & 31 & 23\end{array}\) To calculate the \(10 \%\) trimmed mean, first rank these data values in increasing order; then drop \(10 \%\) of the smallest values and \(10 \%\) of the largest values. The mean of the remaining \(80 \%\) of the values will give the \(10 \%\) trimmed mean. Note that this data set contains 10 values, and \(10 \%\) of 10 is 1 . Thus, if we drop the smallest value and the largest value from this data set, the mean of the remaining 8 values will be called the \(10 \%\) trimmed mean. Calculate the \(10 \%\) trimmed mean for this data set.

The following data give the prices (in thousands of dollars) of 20 houses sold recently in a city. 1 3 \(\begin{array}{llllllllll}184 & 297 & 365 & 309 & 245 & 387 & 369 & 438 & 195 & 390 \\ 323 & 578 & 410 & 679 & 307 & 271 & 457 & 795 & 259 & 590\end{array}\) Find the \(20 \%\) trimmed mean for this data set.
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