Unit 1 practice test: Exploring One-Variable Data and Collecting Data

$\text{IQR} = 79 - 63 = 16$; fences at $63 - 1.5(16) = 39$ and $79 + 1.5(16) = 103$. The min and max are inside, so the whiskers reach 41 and 98.

• Yes: the score of 41 is far below the median of 72: Far below the median is not enough; 41 is above the lower fence of 39.
• Yes: the score of 98 is far above the third quartile of 79: 98 is below the upper fence of 103, so it is not an outlier.
• Correct: No: the fences are 39 and 103, and every score is between them: Right: $\text{IQR} = 16$, so the fences are $63 - 24 = 39$ and $79 + 24 = 103$. The minimum 41 and maximum 98 are inside.
• It is impossible to decide without the mean and standard deviation: The $1.5 \times \text{IQR}$ rule needs only $Q_1$ and $Q_3$, and the extremes.

The box spans the middle 50% of the data: about $0.5 \times 120 = 60$ students.

• About 30: A quarter lies in each whisker section; the box holds two quarters.
• About 90: About three quarters scored 79 or less; between 63 and 79 is the middle half.
• Correct: About 60: Right: the box, from $Q_1$ to $Q_3$, holds the middle half of the data: about half of 120.
• About 16: 16 points is the width of the box, not a count of students.

85 lies between $Q_3 = 79$ and the maximum 98, the top quarter of the scores.

• Correct: It is in the top quarter of the scores: Right: 85 is above $Q_3 = 79$, so it is higher than about 75% of the scores.
• It is an outlier: Values above $Q_3$ are ordinary parts of the data; 85 is well inside the upper fence of 103.
• It is near the median, since it is near the middle of 41 and 98: The median is 72. The middle of the min and max is not the median.
• It is in the middle half of the scores, inside the box: The box ends at 79; 85 is beyond it.

Each quarter holds the same number of values; a wider section means those values are more spread out. Here 63 − 41 = 22 and 98 − 79 = 19.

• The top quarter has more students than the lowest quarter: Each quarter holds about 30 students; the quarters differ in width, not in count.
• The two quarters are equally spread, as each holds a quarter of the scores: Equal counts do not mean equal widths: 22 points against 19.
• The top quarter is more spread out, since it reaches the highest score: Reaching the maximum is not the same as being wide: 79 to 98 is 19 points.
• Correct: The lowest quarter (41 to 63) is more spread out than the top quarter (79 to 98): Right: 22 points against 19 points, so the lowest quarter of scores is more spread out.

Two peaks with a gap between them make the distribution bimodal.

• Skewed right: A long tail would be a few large values trailing off; here there is a whole second group.
• Uniform: A uniform shape has bars of about equal height everywhere; this histogram has an empty stretch in the middle.
• Correct: Bimodal: Right: there are two separate peaks, one near 155 cm and one near 190 cm.
• Unimodal: The two sides match, but there are two peaks and almost no one near 175.

Two clusters usually point to two kinds of individuals in one data set. Here that is two sports.

• Correct: The athletes come from two sports with very different typical heights: Right: a bimodal distribution often signals that two groups have been mixed together, such as gymnasts and basketball players.
• Some athletes were measured in inches instead of centimeters: Heights in inches would be near 60 to 80, far below the values here.
• The heights were rounded to the nearest 5 centimeters: Rounding would change bars a little; it would not create two groups 30 cm apart.
• Tall athletes are more likely to be chosen for the camp: That would push the data up, not split it into two separate groups.

The empty bins between about 165 and 180 cm are a gap between two clusters.

• An outlier at 202 centimeters, the tallest athlete in the camp: 202 is close to the other tall athletes, inside the upper cluster.
• An outlier at 148 centimeters, the shortest athlete in the camp: 148 sits with the lower cluster of heights.
• A uniform stretch: The stretch from 165 to 180 is empty, so the heights are far from uniform.
• Correct: A gap: no athlete is between 165 and 180 centimeters: Right: the empty bins between the two clusters form a gap.

In a bimodal distribution a single center can fall in a gap where there are no values. Describe each cluster separately.

• 175 centimeters is too tall to be a typical height for an athlete: The problem is not the size of 175 but that no athlete is near it.
• Correct: Hardly anyone is near 175; the heights form two groups: Right: a single center hides the two groups. It is better to describe each group on its own.
• A center is not defined for heights measured in centimeters: Centers can be given in any unit; this one just falls in the gap.
• It should be in inches: The unit does not matter; a center in the gap misleads in any unit.