Question

The number of passes completed by Brett Favre, quarterback for the Green Bay Packers, was recorded for each of the 16 regular season games in the fall of 2006 (www.espn.com). \(^{9}\) $$ \begin{array}{rrrrrr} 15 & 31 & 25 & 22 & 22 & 19 \\ 17 & 28 & 24 & 5 & 22 & 24 \\ 22 & 20 & 26 & 21 & & \end{array} $$ a. Draw a stem and leaf plot to describe the data. b. Calculate the mean and standard deviation for Brett Favre's per game pass completions. c. What proportion of the measurements lie within two standard deviations of the mean?

Short answer

Answer: The proportion of measurements within two standard deviations of the mean is approximately 0.9375 or 93.75%.

Step-by-step solution

Organize the data

First, let's organize the data from the lowest to the highest value: 5, 15, 17, 19, 20, 21, 22, 22, 22, 22, 24, 24, 25, 26, 28, 31

Draw a stem and leaf plot

Now, create a stem and leaf plot from the organized data: Stem | Leaf 0 | 5 1 | 579 2 | 0112223445568 3 | 1

Calculate the mean

To find the mean of Brett Favre's completed passes per game, sum the values and divide by the total number of games (16). Mean = (5+15+17+19+20+21+22+22+22+22+24+24+25+26+28+31)/16 = 343/16 Mean = 21.44 completed passes per game

Calculate the standard deviation

To calculate standard deviation, follow these steps: 1. Find the mean (already calculated): 21.44 2. Subtract the mean from each value and square the result 3. Calculate the average of squared differences (variance) 4. Take the square root of the variance Variance = [(5-21.44)^2 + (15-21.44)^2 + ... + (31-21.44)^2]/16 Variance ≈ 47.90 Standard Deviation = √47.90 ≈ 6.92 passes

Determine the proportion of measurements within two standard deviations

By calculating mean ± 2 standard deviations, we will find the range: Lower Range = Mean - 2*(Standard Deviation) = 21.44 - 2*6.92 ≈ 7.60 Upper Range = Mean + 2*(Standard Deviation) = 21.44 + 2*6.92 ≈ 35.28 Now, we count the total number of measurements within this range: There are 15 out of 16 measurements within two standard deviations of the mean. The proportion of measurements within two standard deviations of the mean is: Proportion = 15/16 ≈ 0.9375 or 93.75%

Key concepts

Stem and Leaf Plot

A stem and leaf plot is a visual method of displaying quantitative data that helps in understanding the distribution and frequency of a data set. It resembles a histogram turned sideways; with stems on one side representing the leading digit(s) and leaves on the other side representing the trailing digit(s). Imagine this like a more detailed bar chart where each bar is split into individual numbers. To create a stem and leaf plot from a dataset, follow these steps:

• Separate each value into a stem (the number's leading digit(s)) and a leaf (the number's trailing digit).
• List the stems in ascending order on the left and attach the corresponding leaves for each stem, also in ascending order.
• It's often helpful to indicate the number of instances a leaf occurs if the data repeats.

For the dataset given in the exercise, with each game's number of passes organized in ascending order, they formed stems from 0 through 3 with the leaves representing the last digits of the number of passes. Stem '2' for example, had several leaves showing how often the twenties range of passes occurred.

Mean Calculation

The mean, often referred to as the average, is a measure of central tendency in a data set that represents the typical value. Calculating the mean involves summing all the values and then dividing by the number of values. The formula for the mean (\bar{x}) is expressed as:

\[\begin{equation}\bar{x} = \dfrac{\sum{x_i}}{n}\end{equation}\]

where represents each value in the dataset and is the total number of values. Applying this to Brett Favre's completed passes, we first summed the number of passes per game and divided by the total number of games, which is 16. This calculation provided the average number of completed passes per game for that season.

Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values are close to the mean of the data set, while a high standard deviation indicates a wide range of values. To calculate the standard deviation:\

• Compute the mean (average) of the data set.
• Subtract the mean from each data point and square the result (the squared difference).
• Find the average of these squared differences (this is the variance).
• Take the square root of the variance to get the standard deviation.

The formula for standard deviation (\sigma) is:\[\begin{equation}\sigma = \sqrt{\dfrac{\sum{(x_i - \bar{x})^2}}{n}}\end{equation}\]
For Brett Favre's data, after finding the mean, each game's passes were subtracted from this mean, squared, and then the average of these squares was taken to find the variance. The square root of this variance gave us the standard deviation, showing us the spread of the number of passes around the average.

Variance

Variance is a statistical measurement of the spread between numbers in a data set. More precisely, it measures how far each number in the set is from the mean and thus from every other number in the set. The formula to calculate variance (\sigma^2) is similar to that of the standard deviation, but without taking the square root at the end. Variance is calculated as follows:\

\[\begin{equation}\sigma^2 = \dfrac{\sum{(x_i - \bar{x})^2}}{n}\end{equation}\]
The variance provides a squared value which can be difficult to apply directly to the initial data as they are not in the same units. That's why the square root is taken to return to the original unit of measurement, transforming variance back into standard deviation. Brett Favre's pass completions had a variance of approximately 47.90, indicating the degree to which individual games deviated from the mean performance.

Probability and Statistics

Probability and statistics deal with the study of randomness and uncertainty. These mathematical fields provide tools for analyzing, interpreting, and making predictions based on data. Probability is about modeling and measuring uncertainty: it uses models and theories to determine the likelihood of certain events occurring. Statistics, on the other hand, pertains to the collection, analysis, interpretation, and presentation of masses of numerical data. It involves using probability to make inferences about a population from a sample.

In the context of the exercise, probability theory is employed to ascertain the likelihood of Brett Favre's game performances falling within a certain range of values. Specifically, when we say 93.75% of the games fall within two standard deviations of the mean, we're using statistical knowledge to draw conclusions about his consistency over the season. These kinds of insights help coaches, players, and fans understand performance levels and can inform future strategies and expectations.