91Ó°ÊÓ

A sample of 10 NCAA college basketball game scores provided the following data \((U S A\) Today, January 26,2004 ). $$\begin{array}{lclcr} & & & & \text { Winning } \\ \text { Winning Team } & \text { Points } & \text { Losing Team } & \text { Points } & \text { Margin } \\ \text { Arizona } & 90 & \text { Oregon } & 66 & 24 \\ \text { Duke } & 85 & \text { Georgetown } & 66 & 19 \\ \text { Florida State } & 75 & \text { Wake Forest } & 70 & 5 \\ \text { Kansas } & 78 & \text { Colorado } & 57 & 21 \\ \text { Kentucky } & 71 & \text { Notre Dame } & 63 & 8 \\ \text { Louisville } & 65 & \text { Tennessee } & 62 & 3 \\ \text { Oklahoma State } & 72 & \text { Texas } & 66 & 6\end{array}$$ $$\begin{array}{lccc} \text { Winning Team } & \text { Points } & \text { Losing Team } & \text { Points } & \text { Winning Margin } \\ \text { Purdue } & 76 & \text { Michigan State } & 70 & 6 \\ \text { Stanford } & 77 & \text { Southern Cal } & 67 & 10 \\ \text { Wisconsin } & 76 & \text { Illinois } & 56 & 20 \end{array}$$ a. Compute the mean and standard deviation for the points scored by the winning team. b. Assume that the points scored by the winning teams for all NCAA games follow a bell-shaped distribution. Using the mean and standard deviation found in part (a), estimate the percentage of all NCAA games in which the winning team scores 84 or more points. Estimate the percentage of NCAA games in which the winning team scores more than 90 points. c. Compute the mean and standard deviation for the winning margin. Do the data contain outliers? Explain.

Step by step solution

01

Calculate Mean for Winning Points

First, list the points scored by each winning team: 90, 85, 75, 78, 71, 65, 72, 76, 77, 76. Sum these scores: \(90 + 85 + 75 + 78 + 71 + 65 + 72 + 76 + 77 + 76 = 765\). Divide by the number of games (10) to find the mean: \(\frac{765}{10} = 76.5\).

02

Calculate Standard Deviation for Winning Points

Subtract the mean (76.5) from each winning score, square the result, and sum the squares: \((90 - 76.5)^2 + (85 - 76.5)^2 + \ldots + (76 - 76.5)^2 = 292.5\). Divide by the number of games minus one (9): \(\frac{292.5}{9} = 32.5\). Take the square root to find the standard deviation: \(\sqrt{32.5} \approx 5.7\).

03

Estimate Percentage for Scores 84+ Using Empirical Rule

For a bell-shaped distribution, calculate the z-score for 84: \( z = \frac{84 - 76.5}{5.7} \approx 1.32\). Using the empirical rule and standard normal distribution, approximately 11% of scores are greater than 84 (beyond one standard deviation).

04

Estimate Percentage for Scores 90+

Calculate the z-score for 90: \( z = \frac{90 - 76.5}{5.7} \approx 2.37\). Referring to the normal distribution, scores more than two standard deviations above the mean represent about 1% of the data.

05

Calculate Mean for Winning Margin

List the winning margins: 24, 19, 5, 21, 8, 3, 6, 6, 10, 20. Sum these margins: \(\sum = 122\). Divide by the number of games: \(\frac{122}{10} = 12.2\).

06

Calculate Standard Deviation for Winning Margin

Subtract the mean (12.2) from each margin, square the result, and sum the squares: \((24 - 12.2)^2 + \ldots + (20 - 12.2)^2 = 621.6\). Divide by \(9\) (degrees of freedom): \(\frac{621.6}{9} = 69.07\). Take the square root to find the standard deviation: \(\sqrt{69.07} \approx 8.31\).

07

Determine Outliers in Winning Margin

An outlier typically lies more than 1.5 times the interquartile range (IQR) beyond the first or third quartile. However, with a standard deviation of \(8.31\), any margin beyond \(12.2 \pm 2 \times 8.31\) could suggest scrutiny for outliers. Here, 3 may be an outlier, but further IQR analysis is suggested.

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Key Concepts

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

Mean Calculation

In descriptive statistics, the **mean** is one of the most important concepts. It's essentially the average, giving you an idea of the central value for a data set. Calculating the mean is usually the first step in analyzing data. Here’s how you can do it:

• First, list all the data points. In this case, these are the scores of the winning teams in basketball games: 90, 85, 75, etc.
• Next, add up all the values. The sum of these points is 765.
• Finally, divide the total by the number of observations, which is 10 games in our work-through.

By following these steps, we find that the mean score of the winning team was 76.5 points. This value serves as a helpful reference point for seeing how individual scores compare to the dataset as a whole.

Standard Deviation

**Standard deviation** is a crucial measure of how spread out numbers are in a data set. This statistic helps you understand the variability or dispersion of the data. Here’s a quick guide to understanding and calculating it:

• Subtract the mean from each data point to find the deviation of each point from the average.
• Square each deviation to eliminate negative values, then sum all the squared deviations.
• Divide this total by the number of values minus one (for a sample standard deviation).
• Finally, take the square root of this quotient to arrive at the standard deviation.

For our basketball scores, the standard deviation is approximately 5.7. A smaller standard deviation indicates that the scores are close to the average, while a larger one suggests more variation.

Empirical Rule

The **Empirical Rule** is a handy guideline that applies to bell-shaped, or normal, distributions. It lets you make quick assessments about data spread in relation to its mean and standard deviation:

• 68% of data falls within one standard deviation from the mean.
• 95% is within two standard deviations.
• 99.7% lies within three standard deviations.

With the mean of 76.5 and a standard deviation of 5.7, we calculate z-scores to make predictions: For scores 84 and above, the z-score is 1.32, indicating we’re about one standard deviation above the mean, and roughly 11% of scores exceed this. For scores above 90, the z-score is 2.37, which denotes more than two standard deviations, so only about 1% of scores surpass this.

Outlier Detection

Outliers are unusual values in a data set that differ significantly from other observations. They can skew and mislead the interpretation of data, so detecting them is critical.
One method is using the standard deviation to identify these points:

• Calculate the mean and standard deviation.
• Typically, any value more than two standard deviations from the mean may be considered a potential outlier.
• For deeper analysis, the Interquartile Range (IQR) method can be applied to further investigate outliers.

In our basketball game data, with a standard deviation of 8.31 for winning margins, only a margin significantly far from the average of 12.2, like 3, could be scrutinized as an outlier. Further statistical tests or visualizing the data might be required to confirm this.