Q: Given the following five-number summary for steel roller coasters: minimum $$\ 4.5 \ \text{mph} \ $$, first quartile $$\ 45.25 \ \text{mph} \ $$, median $$\ 56.51294 \ \text{mph} \ $$, third quartile $$\ 67 \ \text{mph} \ $$, maximum $$\ 149.1 \ \text{mph} \ $$, what is the interquartile range (IQR)?

$$\ 67 - 45.25 = 21.75 \ \text{mph} \ $$

Question 1

Which of the following best describes the distribution of roller coaster speeds around the world, as shown in the histogram?

Accepted Answer

It is right-skewed with a single large peak around 60-70 mph.

Question 2

Given the summary statistics for roller coaster speeds, which measure of central tendency is most appropriate for describing the typical speed, and why?

Accepted Answer

Median, because the data is skewed and contains outliers.

Question 3

If a roller coaster has a speed of 110 mph, how many standard deviations above the mean is this speed? Use the mean $$\ 56.82 \ 	ext{mph} \ $$ and standard deviation $$\ 18.3875 \ 	ext{mph} \ .$$

Accepted Answer

$$\ \frac{110 - 56.82}{18.3875} \approx 2.89 \ $$

Question 4

Based on the boxplot comparing steel and wood roller coaster tracks, which statement is most accurate?

Accepted Answer

Steel tracks have a much larger range and more outliers than wood tracks.

Question 5

If you want to compare the spread of speeds for steel and wood roller coasters, which statistical measure would be most informative?

Accepted Answer

Standard deviation

Question 6

A new steel roller coaster is built with a speed of 140 mph. Based on the steel track summary statistics, is this speed considered an outlier?

Accepted Answer

Yes, because it is much higher than the third quartile and maximum for wood tracks.

Question 7

Which region in the US spends the least amount of time on physical activity, according to the boxplot?

Accepted Answer

South

Question 8

If the median physical activity time for the West region is 55 minutes and for the South is 45 minutes, what does this indicate about the distribution of activity times?

Accepted Answer

The West region has a higher central tendency for physical activity than the South.

Question 9

Given the following five-number summary for steel roller coasters: minimum $$\ 4.5 \ 	ext{mph} \ $$, first quartile $$\ 45.25 \ 	ext{mph} \ $$, median $$\ 56.51294 \ 	ext{mph} \ $$, third quartile $$\ 67 \ 	ext{mph} \ $$, maximum $$\ 149.1 \ 	ext{mph} \ $$, what is the interquartile range (IQR)?

Accepted Answer

$$\ 67 - 45.25 = 21.75 \ 	ext{mph} \ $$

Question 10

If a physical activity time in the Midwest region is 60 minutes, and the mean is 45.46923 minutes with a standard deviation of 4.806833 minutes, what is the z-score for this value?

Accepted Answer

$$\ \frac{60 - 45.46923}{4.806833} \approx 3.03 \ $$

Question 11

Why might the mean not be the best measure of central tendency for the roller coaster speed data?

Accepted Answer

Because the data is skewed and contains outliers.

Question 12

If you want to compare the variability of physical activity times between the Northeast and Midwest regions, which statistic should you use?

Accepted Answer

Standard deviation

Question 13

A roller coaster has a speed equal to the third quartile for steel tracks. What is this speed?

Accepted Answer

$$\ 67 \ 	ext{mph} \ $$

Question 14

Which of the following statements is supported by the boxplot of physical activity levels by US region?

Accepted Answer

The West region has the highest median and the largest spread in activity times.

Question 15

If the standard deviation of wood track roller coaster speeds is $$\ 9.028098 \ 	ext{mph} \ $$ and the mean is $$\ 58.98000 \ 	ext{mph} \ $$, what percentage of wood track roller coasters are expected to have speeds within one standard deviation of the mean, assuming a normal distribution?

Accepted Answer

Approximately 68%

Question 16

Which summary statistic is most resistant to the effect of outliers in the roller coaster speed data?

Accepted Answer

Median

Question 17

If you were to use the five-number summary to describe the spread of roller coaster speeds, which values would you include?

Accepted Answer

Minimum, first quartile, median, third quartile, maximum

Question 18

A physical activity time in the South region is 60 minutes. If the mean is 51.47273 minutes and the standard deviation is 3.633205 minutes, is this value an outlier?

Accepted Answer

Yes, because it is more than two standard deviations above the mean.

Question 19

If the range of speeds for wood track roller coasters is $$\ 74 - 40 = 34 \ 	ext{mph} \ $$, what does this tell you about the variability of wood track speeds compared to steel tracks?

Accepted Answer

Wood tracks have less variability in speed than steel tracks.

Statistical Analysis of Roller Coaster Speeds and Physical Activity Levels

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