Chapter Three
1. Using the following data for high school track athlete’s times (in seconds) for sprinting the 100-m dash, calculate the following:
a. Mean
b. Median
c. Mode
d. Lower Quartile
e. Upper Quartile
f. 80th Percentile
g. What percentile would a racer be if they finished with a time of 17.5 seconds?
h. What percentile is the racer who finished a time of 13.9 seconds?
14.8 16.3 13.9
16.2 12.7 14.1
14.4 13.7 15.2
14.1 17.0 13.4
18.8 14.4 14.3
13.1 11.5 15.8
13.2 15.6 18.1
2. The grading for this course is made up of the following components:
Professionalism 5%
Problem Sets 18%
Project 25%
Midterms 22%
Final 30%
Calculate the final grade of a student who had the following scores:
Professionalism 10/10
Problem sets 53/60
Project 193/200
Midterms 17/90
Final 39/75
Next, calculate what their grade would be if each component had equal weighting. Why wouldn’t a professor use equal weighting?
3. The following data is on the number of books checked out at the Fraser Valley Regional Library’s Langley Branch over the course of 12 months.
a. Assume this data is a sample. Compute the range and standard deviation. Use the “long way” of calculating standard deviation by breaking it into its constituent parts like in the demonstration spreadsheet.
b. Now assume this data is a population (i.e. this branch has only been open for a year). Compute the standard deviation.
c. Indicate why these numbers are different. (I.e. why are the formulae for samples and populations different? What purpose does it serve?)
5,175 5,781 5,384
4,139 4,531 4,871
5.005 5,233 5,670
6.921 4,197 4,188
4. Use the following two distributions.
Distribution A Distribution B
µ = 45,600 µ = 33.4
ó = 6,333 ó = 4.05
a. Compute the coefficient of variation for each distribution.
b. If a value drawn from A is 50,000 and a value drawn from B is 40, convert each value into a z-score and indicate which is relatively closer to its respective mean.
5. In the early 1970s, a Vessel Sanitation Program (VSP) was started for the cruise ship industry because several diseases had broken out on cruise ships. The VSP protects the health of passengers and crew by scoring each ship based on a 100-point scale for sanitation. A boat that achieves a score of 86 or higher is considered to have satisfactory sanitation. (Data from a recent inspection are provided in an accompanying file.)
a. Calculate the mean, standard deviation, median and inter-quartile range
b. Look at the distribution of score visually. Would the Empirical Rule or Chebyshev’s theorem be preferred to describe this data set?
c. If a passenger only wanted to travel on ships that were at the 90th percentile or higher in terms of VSP ratings, what is the lowest sanitation score they would find acceptable?
6. (Based on Case 3.4 in edition 8) AJ Fitness is a new fitness club in town. AJ Reeser bought out an existing health club (known as the Park Centre Club) and conducted a survey of its existing membership of 1,833 to find out their current level of satisfaction. (AJ knows it is a lot cheaper to keep existing customers than trying to attract new ones.) The response rate to the survey was very good for a mail-out survey – 1,214 members returned the survey. AJ hired a business student from the local university to conduct an analysis of the results. When the student asked him for some direction in the project, AJ’s reply was, “That’s what I hired you for. I just want a descriptive analysis of these results. Develop whatever charts, graphs and tables that will help us understand our customers. Also use whatever pertinent numerical measures that will help us.” AJ set up a time next week to meet to discuss the student’s work.
Develop descriptive statistics (both graphical and numerical) that could be used in the report
Chapter Three
1. Using the following data for high school track athlete’s times (in seconds) for sprinting the 100-m dash, we can calculate the following:
a. Mean:
To calculate the mean, you sum up all the data points and divide it by the total number of data points.
Mean = (14.8 + 16.3 + 13.9 + 16.2 + 12.7 + 14.1 + 14.4 + 13.7 + 15.2 + 14.1 + 17.0 + 13.4 + 18.8 + 14.4 + 14.3 + 13.1 + 11.5 + 15.6 + 18.1) / 19
Mean = 279.6 / 19
Mean ≈ 14.71 seconds
b. Median:
To calculate the median, you arrange the data points in ascending order and find the middle value. If there is an even number of data points, you take the average of the two middle values.
Median = 14.1
c. Mode:
The mode is the data point that appears most frequently.
Mode = 14.1
d. Lower Quartile:
The lower quartile divides the lower 25% of the data from the upper 75%.
Lower Quartile = 13.9
e. Upper Quartile:
The upper quartile divides the lower 75% of the data from the upper 25%.
Upper Quartile = 15.6
f. 80th Percentile:
The 80th percentile represents the data point below which 80% of the data falls.
80th Percentile ≈ 15.8
g. Percentile for 17.5 seconds:
To calculate the percentile, you need to determine the position (rank) of the data point within the data set.
Percentile = (Number of data points below 17.5) / Total number of data points * 100
Percentile = (14/19) * 100
Percentile ≈ 73.68%
h. Percentile for 13.9 seconds:
Percentile = (Number of data points below 13.9) / Total number of data points * 100
Percentile = (8/19) * 100
Percentile ≈ 42.11%
2. The grading for this course is as follows:
Professionalism: 5%
Problem Sets: 18%
Project: 25%
Midterms: 22%
Final: 30%
a. Calculate the final grade for a student who had the following scores:
Professionalism: 10/10
Problem Sets: 53/60
Project: 193/200
Midterms: 17/90
Final: 39/75
Final Grade = (10/10) * 0.05 + (53/60) * 0.18 + (193/200) * 0.25 + (17/90) * 0.22 + (39/75) * 0.30
Final Grade ≈ 0.05 + 0.159 + 0.24125 + 0.0414444 + 0.156
Final Grade ≈ 0.6476944
b. Calculate the grade if each component had equal weighting:
If each component had equal weighting, we would assign each component a weight of 1/5 since there are 5 components.
Final Grade (Equal Weighting) = (10/10) * 1/5 + (53/60) * 1/5 + (193/200) * 1/5 + (17/90) * 1/5 + (39/75) * 1/5
Final Grade (Equal Weighting) ≈ 0.2 + 0.1768 + 0.193 + 0.0377778 + 0.208
Final Grade (Equal Weighting) ≈ 0.8155778
Professors typically do not use equal weighting because it does not effectively capture the importance of each component. Some components may carry more weight or be more indicative of a student's overall performance in the course.
1. To calculate the various statistical measures for the high school track athlete's times, follow these steps:
a. Mean: Add up all the times and divide by the total number of times.
b. Median: Arrange the times in ascending order and find the middle value. If there's an even number of times, take the average of the two middle values.
c. Mode: Identify the time that appears most frequently.
d. Lower Quartile: Arrange the times in ascending order and find the median of the lower half.
e. Upper Quartile: Arrange the times in ascending order and find the median of the upper half.
f. 80th Percentile: Arrange the times in ascending order and find the value that separates the bottom 80% from the top 20%.
g. To determine the percentile for a racer finishing with a time of 17.5 seconds, arrange the times in ascending order and count the number of times that are slower than 17.5 seconds. Divide this count by the total number of times and multiply by 100.
h. To determine the percentile for a racer finishing with a time of 13.9 seconds, follow the same process as in step g.
2. To calculate the final grade of a student given the weightings for each component, multiply each score by its respective weighting percentage, then sum up the products.
To calculate the grade with equal weighting, assign equal weight to each component (20% for each, since there are five components) and calculate the final grade using the same method.
Professors might not use equal weighting because different components may require more or less effort, or reflect different levels of importance in assessing a student's overall performance in the course.
3. a. To calculate the range, subtract the smallest value from the largest value in the data set.
To calculate the standard deviation, follow these steps:
- Find the mean of the data set
- Subtract the mean from each value in the data set
- Square each result
- Find the mean of the squared results
- Take the square root of the mean from the previous step
b. If the data is considered a population, the calculation for standard deviation does not involve dividing by (n-1), but by n instead.
c. The formulas for calculating standard deviation differ for samples and populations because using different formulas adjusts for the differing amount of uncertainty in estimating the population parameter (mean) based on a sample. The (n-1) correction in the sample formula accounts for the fact that the sample mean is used to estimate the population mean, introducing some level of uncertainty.
4. a. To compute the coefficient of variation for each distribution, divide the standard deviation by the mean and multiply by 100.
b. To convert a value into a z-score, subtract the mean from the value and divide by the standard deviation. Compare the absolute values of the z-scores to determine which value is relatively closer to its respective mean.
5. a. To calculate the mean, add up all the scores and divide by the total number of scores.
To calculate the standard deviation, follow the same steps as in question 3a.
To calculate the median, arrange the scores in ascending order and find the middle value. If there's an even number of scores, take the average of the two middle values.
To calculate the inter-quartile range, subtract the lower quartile from the upper quartile.
b. To determine which rule to prefer, consider the shape of the distribution. If the distribution is approximately bell-shaped and symmetric, the Empirical Rule (also known as the 68-95-99.7 rule) can be used. If the distribution is not bell-shaped or not symmetric, Chebyshev's theorem can be used.
c. To find the lowest sanitation score acceptable for the 90th percentile, sort the scores in descending order and find the score that corresponds to the 90th percentile.
6. To develop descriptive statistics for AJ Fitness' survey results, consider generating the following:
- Charts: Bar charts or pie charts to visualize customer satisfaction levels for different aspects.
- Graphs: Line graphs or scatter plots to show trends or correlations between variables.
- Tables: Tabulate customer responses for each satisfaction aspect.
- Numerical measures: Calculate mean, median, and standard deviation for customer satisfaction ratings to provide an overall understanding of satisfaction levels.
These descriptive statistics can help AJ understand customer satisfaction patterns and identify areas for improvement.