Chapter 2 Questions
Question 1
Consider the following graph. The idea is that crime is rising, as the caption notes with preparing to deal with increased crime.
(a) Does the graph show an increase in crime over time?
(b) What elements are not particularly good for making the case that crime is rising?
Answer
(a) The timeline on the x axis is reversed, so time is increasing from right to left instead of the usual left to right. Hence the graph actually shows a general decrease in crime, although the last period is higher than the second last period shown it is still smaller than earlier. In looking at time plots you need to always look at the axes to be able to make sense of them.
(b) Reversing the timeline is certainly not good, especially as it is designed to give the wrong idea. More subtly we really do not know what the numbers actually are. Are they actual crime numbers? It does not look like it (what woudl 52% crime be?). It seems that what they are reporting is the proportion of adults that THINK that crime is a very big problem. Which is a whole different thing. In looking at any plot you need to think about what the data really is. A good question to always ask is 'how can they measure that?'.Question 2
A tech startup had the following monthly revenues (in tens of thousands of dollars) over the last 5 months:
| Month 1 | Month 2 | Month 3 | Month 4 | Month 5 |
|---|---|---|---|---|
| 1.2 | 0.8 | 2.0 | 1.6 | 1.4 |
(a) What is the median level of revenue over these months?
(b) What is the mean level of revenue over these months?
(c) Does the relationship between the mean and median tell us anything about the distribution of revenue for this example?
Answer
(a) Ordering these values, we have {0.8, 1.2, 1.4, 1.6, 2.0}, so the median is 1.4.
(b) The sample mean is:
x̄ = (1/5) × (1.2 + 0.8 + 2.0 + 1.6 + 1.4) = (1/5) × (7.0) = 1.4
(c) The mean and median are equal (1.4). While equality between the mean and median often suggests a symmetric distribution, equality alone does not guarantee symmetry. However, inspecting the ordered data points relative to the center ({1.4 - 0.6, 1.4 - 0.2, 1.4, 1.4 + 0.2, 1.4 + 0.6}) shows that this specific dataset is perfectly symmetric.
Question 3
A small retail store recorded its daily customer count over 5 consecutive days:
| Day 1 | Day 2 | Day 3 | Day 4 | Day 5 |
|---|---|---|---|---|
| 12 | 15 | 40 | 18 | 15 |
(a) What is the median customer count over these days?
(b) What is the mean customer count over these days?
(c) Does the relationship between the mean and median tell us anything about the distribution of customer counts for this example?
Answer
(a) Ordering these values, we have {12, 15, 15, 18, 40}, so the median is 15.
(b) The sample mean is:
x̄ = (1/5) × (12 + 15 + 40 + 18 + 15) = (1/5) × (100) = 20
(c) Here, the mean (20) is greater than the median (15). This indicates that the distribution is right-skewed, pulled upward by an unusually high data value (the outlier of 40 on Day 3).
Question 4
A local coffee shop tracked the daily rainfall (in millimeters) recorded during a 5-day storm period:
| Day 1 | Day 2 | Day 3 | Day 4 | Day 5 |
|---|---|---|---|---|
| 2.0 | 6.0 | 10.0 | 9.0 | 8.0 |
(a) What is the median level of rainfall over these days?
(b) What is the mean level of rainfall over these days?
(c) Does the relationship between the mean and median tell us anything about the distribution of rainfall for this example?
Answer
(a) Ordering these values, we have {2.0, 6.0, 8.0, 9.0, 10.0}, so the median is 8.0.
(b) The sample mean is:
x̄ = (1/5) × (2.0 + 6.0 + 10.0 + 9.0 + 8.0) = (1/5) × (35.0) = 7.0
(c) Here, the mean (7.0) is smaller than the median (8.0). This indicates that the distribution is left-skewed, pulled downward by the lower measurement of 2.0 mm on Day 1.
Question 4
Does study time guarantee higher exam scores? Consider the relationship between study hours (on the x-axis) and final test scores (on the y-axis) in an introductory statistics course.
(a) Is the correlation between study hours and test scores positive or negative?
(b) Does the relationship seem like it is strong or weak?
Answer
(a) The correlation is positive, as overall higher study hours tend to correspond with higher exam scores.
(b) The relationship is quite weak. The data points are broadly scattered, with noticeable outliers such as students who scored highly with minimal study or scored poorly despite long study hours.
Question 5
Does a larger engine size lead to worse fuel efficiency? Consider the relationship between engine displacement in liters (on the x-axis) and miles per gallon (on the y-axis) for a sample of modern sedans.
(a) Is the correlation between engine size and fuel efficiency positive or negative?
(b) Does the relationship seem like it is strong or weak?
Answer
(a) The correlation is negative, as points go down to the right, showing that larger engine displacements correspond with lower MPG.
(b) The relationship is strong. The points tightly follow a distinct linear trajectory with very little dispersion.
Copyright © Graham Elliott
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