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Summarizing Soft Drink Purchases and Audit Times
Published on March 14, 2025
Statistics
The solution provides step-by-step calculations for relative and percent frequency distributions, bar graphs, pie charts, frequency distributions, cumulative frequency distributions, histograms, frequency polygons, and measures of central tendency (mean, median, mode) and dispersion (variance, standard deviation) for sets of data related to soft drink purchases and audit times.
Question
Answer
Here are the step-by-step solutions to the math problems:
Question 4:
a) Relative and percent frequency distributions:
First, we count the frequency of each soft drink from the given data:
- Coca-Cola: 19
- Diet Coke: 8
- Dr. Pepper: 5
- Pepsi: 13
- Sprite: 5
Total number of purchases = 50
Now, we calculate the relative frequency and percent frequency for each soft drink:
| Soft Drink | Frequency | Relative Frequency (
|
Percent Frequency (Relative Frequency
|
|---|---|---|---|
| Coca-Cola | 19 |
|
|
| Diet Coke | 8 |
|
|
| Dr. Pepper | 5 |
|
|
| Pepsi | 13 |
|
|
| Sprite | 5 |
|
|
| Total | 50 | 1.00 | 100% |
b) Bar graph and Pie chart:
-
Bar graph: A bar graph would have soft drinks on the x-axis and relative frequency on the y-axis. The height of each bar corresponds to the relative frequency calculated above.
-
Pie chart: A pie chart represents the percentage distribution. To determine the angle for each segment, we multiply the percent frequency by 3.6 degrees (since a circle has 360 degrees).
- Coca-Cola:
- Diet Coke:
- Dr. Pepper:
- Pepsi:
- Sprite:
Sum of angles: - Coca-Cola:
c) Three most popular soft drinks:
Based on the frequencies, the three most popular soft drinks are:
- Coca-Cola (Frequency: 19)
- Pepsi (Frequency: 13)
- Diet Coke (Frequency: 8)
Question 5:
a) Number of classes, class interval, frequency distribution:
-
Number of classes: Using the rule of thumb, number of classes
. We can choose 5 classes. -
Class interval: Range =
. Class interval . We can use a class interval of 5 for simplicity. -
Classes and Frequency Distribution: Starting from the minimum value 12, with a class interval of 5:
Class Class Boundaries Frequency Relative Frequency Percent Frequency 12-16 11.5 - 16.5 5 17-21 16.5 - 21.5 7 22-26 21.5 - 26.5 5 27-31 26.5 - 31.5 2 32-36 31.5 - 36.5 1 Total 20 1.00 100%
b) Cumulative frequency distribution:
| Class Boundaries | Frequency | Cumulative Frequency |
|---|---|---|
| Less than 11.5 | 0 | 0 |
| Less than 16.5 | 5 | 5 |
| Less than 21.5 | 7 | 12 |
| Less than 26.5 | 5 | 17 |
| Less than 31.5 | 2 | 19 |
| Less than 36.5 | 1 | 20 |
c) Histogram and comment on shape:
A histogram would be drawn with class boundaries on the x-axis and frequency on the y-axis. Observing the frequencies (5, 7, 5, 2, 1), the distribution is slightly skewed to the right (positively skewed) because the tail is longer on the right side. The distribution is unimodal as there is one peak in the frequencies.
d) Frequency polygon and interpret audit times:
A frequency polygon is formed by connecting the midpoints of the bars in a histogram. The midpoints of the classes are: (12+16)/2=14, (17+21)/2=19, (22+26)/2=24, (27+31)/2=29, (32+36)/2=34. The frequency polygon would show the trend of audit times. The audit times are mostly concentrated in the 17-21 days range, with fewer audits taking longer or shorter times. The polygon visually represents the distribution and shape, reinforcing the observation of a slightly right-skewed distribution.
Question 7:
a) Grouped frequency distribution:
| Class Interval | Class Midpoint | Frequency |
|---|---|---|
| 42.0-43.9 | 42.95 | 4 |
| 44.0-45.9 | 44.95 | 12 |
| 46.0-47.9 | 46.95 | 10 |
| 48.0-49.9 | 48.95 | 10 |
| 50.0-51.9 | 50.95 | 8 |
| 52.0-53.9 | 52.95 | 6 |
| Total | 50 |
b) Mean, median, mode, variance, and standard deviation:
-
Mean:
-
Median:
Median position =. Cumulative frequency: - Up to 43.9: 4
- Up to 45.9: 4+12 = 16
- Up to 47.9: 16+10 = 26. Median class is 46.0-47.9.
, , ,
-
Mode:
Modal class is 44.0-45.9 with the highest frequency (12).
, , , , -
Variance:
-
Standard deviation:
c) Number of students who took more than 50 minutes:
Students who took more than 50 minutes are in the classes 50.0-51.9 and 52.0-53.9.
Number of students = Frequency of (50.0-51.9) + Frequency of (52.0-53.9) =
Number of students = Frequency of (50.0-51.9) + Frequency of (52.0-53.9) =
Final Answer: The final answer is