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Chapter 6: Problem 9

The January 1990 issue of Arizona Trend contains a supplement describing the 12 "best" golf courses in the state. The yardages (lengths) of these courses are as follows: 6981 , 7099,6930,6992,7518,7100,6935,7518,7013,6800,7041 and \(6890 .\) Calculate the sample mean and sample standard deviation. Construct a dot diagram of the data.

Short Answer

Expert verified
Sample mean is approximately 7059.75 and standard deviation is approximately 208.48.

Step by step solution

01

Calculate the Sample Mean

To calculate the sample mean, sum up all the yardages and divide by the number of golf courses. The yardages are: 6981, 7099, 6930, 6992, 7518, 7100, 6935, 7518, 7013, 6800, 7041, and 6890. First, sum these values:\[ 6981 + 7099 + 6930 + 6992 + 7518 + 7100 + 6935 + 7518 + 7013 + 6800 + 7041 + 6890 = 84717 \]Then, divide by 12 (the number of courses):\[ \text{Sample mean} = \frac{84717}{12} \approx 7059.75 \]
02

Calculate the Sample Standard Deviation

First, find the deviation of each yardage from the mean, square these deviations, and sum them up:\[ (6981 - 7059.75)^2 + (7099 - 7059.75)^2 + \ldots + (6890 - 7059.75)^2 \]This results in:\[ 6176.0625 + 1538.0625 + 1669.0625 + 451.5625 + 210119.0625 + 1617.5625 + 1561.5625 + 210119.0625 + 69.0625 + 6760.0625 + 334.5625 + 28714.5625 = 478130 \]Now divide the sum by (n-1) to get the variance, where n is the number of courses:\[ \text{Variance} = \frac{478130}{11} \approx 43466.36 \]Take the square root of the variance to get the standard deviation:\[ \text{Sample standard deviation} = \sqrt{43466.36} \approx 208.48 \]
03

Construct a Dot Diagram

To create a dot diagram, place a number line that covers the range of the yardages. Then, above each yardage value, place a dot for each corresponding data point. For example, the value 6981 will have one dot and the value 7518 will have two dots. This visually represents the distribution of yardage lengths.

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Key Concepts

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

Sample Mean Calculation
The sample mean is an important concept in statistics, providing an average that represents the dataset as a whole. To calculate the sample mean of a group of numbers, you simply sum all the numbers and divide by how many numbers there are.
In this exercise, to find the sample mean of the yardages of the 12 best golf courses in Arizona, start by adding up all the yardage values:
  • Add the yardages: 6981, 7099, 6930, 6992, 7518, 7100, 6935, 7518, 7013, 6800, 7041, and 6890. The sum comes out to be 84717.
  • The formula for the sample mean is: \( \bar{x} = \frac{\sum x}{n} \), where \( \sum x \) is the total sum of values and \( n \) is the number of values.
  • Divide the total sum, 84717, by the number of courses (12): \( \frac{84717}{12} \approx 7059.75 \).
The sample mean yardage of these golf courses is approximately 7059.75 yards. This average provides a central value around which the other data points cluster.
Sample Standard Deviation Calculation
The sample standard deviation is a measure of how spread out the numbers in your dataset are around the mean. It gives insight into the variability within your data set. A smaller standard deviation means the values are closely clustered around the mean, while a larger one indicates more spread.To find the sample standard deviation of the yardages:
  • First, calculate each yardage's deviation from the mean and square these differences.
  • Using the data, compute the squared deviations: \((6981 - 7059.75)^2, (7099 - 7059.75)^2, \ldots\)
  • The sum of these squared deviations is 478130.
  • Calculate the variance by dividing by (n-1), where \(n = 12\): \( \frac{478130}{11} \approx 43466.36 \).
  • The sample standard deviation is the square root of this variance: \(\sqrt{43466.36} \approx 208.48\).
This sample standard deviation of approximately 208.48 yards tells us there is some variability in the golf course lengths, but they do generally cluster around the mean of 7059.75 yards.
Dot Diagram Construction
A dot diagram, also known as a dot plot, is a simple way to visualize the frequency of data points along a number line. It's a helpful tool for displaying small sets of data to quickly observe patterns or clusters. To construct a dot diagram for the golf course yardages:
  • Draw a horizontal number line that encompasses the range of the yardages from 6800 to 7518.
  • Above each yardage on the line, place a dot for each occurrence of that value in the data set. For example, since the value 7518 appears twice, it gets two dots stacked vertically.
  • Continue placing dots over each yardage value from your list: 6981, 7099, 6930, 6992, 7518, etc.
This visual representation immediately shows the distribution and frequency of the golf yardages, making it easier to spot which yardages are most common and detect any outliers or clusters.

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Most popular questions from this chapter

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