91Ó°ÊÓ

In the game of golf, distance control is just as important as how far a player hits the ball. Michael went to the driving range with his range finder and hit 75 golf balls with his pitching wedge and measured the distance each ball traveled (in yards). He obtained the following data: $$ \begin{array}{rrrrrrrrr} \hline 100 & 97 & 101 & 101 & 103 & 100 & 99 & 100 & 100 \\ \hline 104 & 100 & 101 & 98 & 100 & 99 & 99 & 97 & 101 \\ \hline 104 & 99 & 101 & 101 & 101 & 100 & 96 & 99 & 99 \\ \hline 98 & 94 & 98 & 107 & 98 & 100 & 98 & 103 & 100 \\ \hline 98 & 94 & 104 & 104 & 98 & 101 & 99 & 97 & 103 \\ \hline 102 & 101 & 101 & 100 & 95 & 104 & 99 & 102 & 95 \\ \hline 99 & 102 & 103 & 97 & 101 & 102 & 96 & 102 & 99 \\ \hline 96 & 108 & 103 & 100 & 95 & 101 & 103 & 105 & 100 \\ \hline 94 & 99 & 95 & & & & & & \\ \hline \end{array} $$ (a) Use statistical software to construct a relative frequency histogram. Comment on the shape of the distribution. Draw a normal density curve on the relative frequency histogram. (b) Do you think the normal density curve accurately describes the distance Michael hits with a pitching wedge? Why?

Step by step solution

01

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Organize the Data: Input the given data set into statistical software such as Excel, R, or Python.

02

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Plot the Histogram: Use statistical software to create a relative frequency histogram of the data.

03

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Draw a Normal Density Curve: Overlay a normal density curve on top of the histogram created in Step 2.

04

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Analyze the Shape of Distribution: Comment on the shape of the histogram. For example, assess whether it is symmetric, skewed, have any outliers, etc.

05

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Judge Normality: Determine if the normal density curve accurately describes the data by assessing the fit of the curve to the histogram.

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Key Concepts

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

Histogram

A histogram is a graphical representation of data that shows the frequency of different intervals (or bins). To create a histogram, you divide your data into these bins and then count how many data points fall into each bin.

You can think of a histogram as a bar chart where each bar represents a range of values rather than a single value.
For example, if you wanted to visualize Michael's golf ball distances, you might have bins for ranges like 90-94 yards, 95-99 yards, etc.

The height of each bar tells you how many golf balls landed in that range. If you use a relative frequency histogram, the height of each bar will represent the proportion of the total data that falls into each bin. This makes it easier to compare datasets of different sizes.

Normal Distribution

A normal distribution, often called a bell curve, is a probability distribution that is symmetric around its mean. This means most of the data points are close to the mean, with fewer points farther away.

In the context of Michael's golf ball distances, if his hitting distances follow a normal distribution, most of his shots would cluster around the average distance.

The key features of a normal distribution are:

• Symmetry: The left and right sides of the graph are mirror images.
• Mean, Median, Mode: All are located at the center of the distribution.
• 68-95-99.7 Rule: Approximately 68% of data falls within 1 standard deviation, 95% within 2 standard deviations, and 99.7% within 3 standard deviations.

You can check if Michael's data follows this pattern by overlaying a normal density curve on his histogram and seeing how closely the data follows this curve.

Data Analysis

Data analysis involves inspecting, cleaning, and modeling data with the goal of discovering useful information. When Michael collects data on his golf ball distances, he is engaging in data analysis to understand his performance.

First, he would organize his data, ensuring that it is clean and complete. Then, he would visualize his data using tools like histograms. By doing this, Michael can spot patterns or outliers.

Analyzing the histogram, Michael can interpret the shape of the distribution. If the distribution is roughly symmetrical and forms a bell shape, it may indicate that his performance follows a normal distribution.

This analysis helps Michael understand his hitting consistency and make necessary adjustments to improve his game.

Relative Frequency

Relative frequency is a measure of how often a particular value occurs in a dataset relative to the total number of observations.

It is calculated by dividing the frequency of a specific value by the total number of observations.

For example, if Michael hits 10 golf balls out of 75 at a distance of 100 yards, the relative frequency for 100 yards is 10/75 or approximately 0.1333.

Using relative frequencies rather than absolute frequencies can help you compare different data sets more easily. A relative frequency histogram shows the proportion of data points in each bin, which is helpful when you want to make comparisons across different datasets.

When Michael creates a relative frequency histogram, he can readily see which distances are most common and how they compare to each other proportionally.