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Chapter 3: Problem 84

The prices of all college textbooks follow a bell-shaped distribution with a mean of \(\$ 105\) and a standard deviation of \(\$ 20\). a. Using the empirical rule, find the percentage of all college textbooks with their prices between i. \(\$ 85\) and \(\$ 125\) ii. \(\$ 65\) and \(\$ 145\) \({ }^{*} \mathrm{~b}\). Using the empirical rule, find the interval that contains the prices of \(99.7 \%\) of college textbooks.

Short Answer

Expert verified
a.(i) 68% of textbooks are priced between $85 and $125, a.(ii) 95% of textbooks are priced between $65 and $145, b. 99.7% of textbooks are priced between $45 and $165.

Step by step solution

01

Identify and Understand Key Information

We know that the average, or mean, price of the textbook is $105, the standard deviation is $20, which is used to measure how spread out the prices are around the average. Therefore, we are dealing with a bell curve or normal distribution which allows us to use the Empirical Rule.
02

Apply the Empirical Rule for a.(i)

The interval $85 to $125 is one standard deviation below and above the mean respectively ($105 - $20 = $85 and $105 + $20 = $125). According to the empirical rule, 68% of all college textbooks are priced between $85 and $125.
03

Apply the Empirical Rule for a.(ii)

The interval $65 to $145 is two standard deviations below and above the mean respectively ($105 - 2*$20 = $65 and $105 + 2*$20 = $145). According to the empirical rule, 95% of all college textbooks are priced between $65 and $145.
04

Apply the Empirical Rule for b

99.7% of the prices lie within three standard deviations of the mean. To find this interval, subtract and then add three times the standard deviation from the mean. This gives an interval of $45 to $165 ($105 - 3*$20 = $45 and $105 + 3*$20 = $165). So, 99.7% of all college textbooks are priced between $45 and $165.

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Key Concepts

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

Bell-Shaped Distribution
A bell-shaped distribution, also known as a Gaussian distribution, forms the classic bell curve that many statistical data sets tend to follow. This shape emerges when data approximates the normal distribution.
Bell-shaped distributions are symmetrical, meaning they have a balanced spread of values on either side of the central peak or the most frequent occurrence (the mean). The "tails" of the distribution taper off on both sides smoothly, indicating the presence of rare, extreme values.
  • The central point represents the mode, which coincides with both the mean and median in a perfectly normal distribution.
  • Bell-shaped distributions offer insights into the variability and predictability of data. The symmetrical shape indicates how data is spread around a central value.
Understanding the bell-shaped distribution is crucial for applying techniques such as the Empirical Rule, which helps summarize data using standard deviations.
Normal Distribution
The normal distribution is a foundational concept in statistics and reflects a continuous probability distribution. Known for its symmetrical bell shape, it is characterized by a specific pattern where most observations cluster around the central peak, tapering off equally towards either side. This distribution can be applied to various real-world phenomena.
For a distribution to be normal, it needs to follow specific properties:
  • It is symmetric around the mean, suggesting an equal number of observations on both sides.
  • The mean, median, and mode are exactly equal, located at the peak.
  • The area under the curve sums up to 1, representing the total probability.
The normal distribution is pivotal because it allows us to use the Empirical Rule, facilitating percentiles and probabilities provided key parameters like the mean and standard deviation.
Standard Deviation
The standard deviation is a measure that expresses the amount of variation or dispersion in a set of values. In simpler terms, it tells us how spread out the numbers in a data set are. If the standard deviation is low, it indicates that the values are tightly clustered around the mean. Conversely, a high standard deviation suggests the values are more spread out.
Standard deviation is crucial in the context of a normal distribution and the Empirical Rule as it defines intervals around the mean.
  • Within one standard deviation鈥� 68% of all observations fall.
  • Two standard deviations capture about 95% of the values.
  • Three standard deviations encompass approximately 99.7% of observations.
Understanding standard deviation is essential for assessing how diverse or uniform data points are relative to the average value, enabling predictions and confidence intervals.
Mean
The mean, commonly known as the average, is a measure of central tendency that proves vital in statistical analyses. It is calculated by summing all values in a data set and dividing by the total number of values. The mean provides a single value summarizing an entire data set, offering a useful point for comparisons and analysis.
In the context of normal distributions:
  • The mean is the peak of the bell curve, where it aligns with the distribution's mode and median.
  • It divides the distribution into two equal halves, being the center around which data points are spread.
  • In a symmetrical distribution, the mean also reflects the point of balance for frequencies.
Understanding the mean is integral to analyze the data's central value and its overall tendencies, making it a cornerstone for further statistical operations and interpretations.

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Most popular questions from this chapter

According to Fair Isaac, "The Median FICO (Credit) Score in the U.S. is 723" (The Credit Scoring Site, 2009). Suppose the following data represent the credit scores of 22 randomly selected loan applicants. \(\begin{array}{lllllllllll}494 & 728 & 468 & 533 & 747 & 639 & 430 & 690 & 604 & 422 & 356 \\ 805 & 749 & 600 & 797 & 702 & 628 & 625 & 617 & 647 & 772 & 572\end{array}\) a. Calculate the values of the three quartiles and the interquartile range. Where does the value 617 fall in relation to these quartiles? b. Find the approximate value of the 30 th percentile. Give a brief interpretation of this percentile. c. Calculate the percentile rank of 533 . Give a brief interpretation of this percentile rank.

Nixon Corporation manufactures computer monitors. The following data are the numbers of computer monitors produced at the company for a sample of 10 days. \(\begin{array}{lllllll}24 & 32 & 27 & 23 & 35 & 33 & 29\end{array}\) 40 23 28 Calculate the mean, median, and mode for these data.

Suppose that there are 150 freshmen engineering majors at a college and each of them will take the same five courses next semester. Four of these courses will be taught in small sections of 25 students each, whereas the fifth course will be taught in one section containing all 150 freshmen. To accommodate all 150 students, there must be six sections of each of the four courses taught in 25 -student sections. Thus, there are 24 classes of 25 students each and one class of 150 students. a. Find the mean size of these 25 classes. b. Find the mean class size from a student's point of view, noting that each student has five classes containing \(25,25,25,25\), and 150 students, respectively. Are the means in parts a and \(\mathrm{b}\) equal? If not, why not?

Each year the faculty at Metro Business College chooses 10 members from the current graduating class that they feel are most likely to succeed. The data below give the current annual incomes (in thousands of dollars) of the 10 members of the class of 2000 who were voted most likely to succeed. \(\begin{array}{llllllllll}59 & 68 & 84 & 78 & 107 & 382 & 56 & 74 & 97 & 60\end{array}\) a. Calculate the mean and median. b. Does this data set contain any outlier(s)? If yes, drop the outlier(s) and recalculate the mean and median. Which of these measures changes by a greater amount when you drop the outlier(s)? c. Is the mean or the median a better summary measure for these data? Explain.

A survey of young people's shopping habits in a small city during the summer months of 2009 showed the following: Shoppers aged 12 to 14 years took an average of 8 shopping trips per month and spent an average of \(\$ 14\) per trip. Shoppers aged 15 to 17 years took an average of 11 trips per month and spent an average of \(\$ 18\) per trip. Assume that this city has 1100 shoppers aged 12 to 14 years and 900 shoppers aged 15 to 17 years. a. Find the total amount spent per month by all these 2000 shoppers in both age groups. b. Find the mean number of shopping trips per person per month for these 2000 shoppers. c. Find the mean amount spent per person per month by shoppers aged 12 to 17 years in this city.
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