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Group Work #2
Answer the following multiple choice questions with the most correct answer.
1. If a histogram of a quantitative variable is multimodal, which of the following must be true?
A. The histogram must be symmetric.
B. The histogram must be normal.
C. The histogram must be skewed left.
D. The histogram must be skewed right.
E. None of the above.
A multi-modal graph does not guarantee any of these four criteria. All it means is that you have
two peaks in your histogram.
2. Which of the following is true about the standard deviation?
A. The standard deviation is used with the mean to describe the spread of your data.
B. The standard deviation is used with the median to describe the spread of the data.
C. The standard deviation is used with the mode to describe the spread of the data.
D. The standard deviation is used with the range to describe the spread of the data.
E. None of the above.
3. Which of the following is true about the quartile system?
A. The first quartile will always be greater than the third quartile.
B. The first quartile is the median of the upper half of the data.
C. The third quartile will always be greater than the first quartile.
D. The third quartile is the median of the upper half of the data.
E. None of the above.
Choice D is correct by definition. You might be tempted to say Choice C, but if all your data points
are the same value, then the first and third quartiles will be the same value.
4. Which of the following is true about the five number summary?
A. It uses the mean as its center value.
B. It uses the minimum to describe the data.
C. It uses the standard deviation to describe the data.
D. It requires that your variable must be qualitative.
E. None of the above.
5. In a symmetric histogram with mean µ = 34.1 and std. deviation σ = 1.3, which of the following is true?
A. x = 35 is an outlier.
B. x = 36 falls within 1 standard deviation of the mean.
C. x = 37 falls within 2 standard deviations of the mean.
D. x = 38 falls within 3 standard deviations of the mean.
E. None of the above are true.
Remember that we use brackets to indicate the range of values, meaning endpoints are included.
6. Which of the following is true in a normal distribution?
A. You will never have outliers.
B. The mean will be the same as the mode.
C. The distribution will be skewed left.
D. Your distribution will be multimodal
E. None of the above are true.
For a normal distribution, the mean, median, and mode are all the same value: the high peak in the middle.
7. For data involving a quantitative variable, which of the following is true?
A. If the graph is symmetric, the best method to use for centrality is the range.
B. If the graph is symmetric, the best method to use for centrality is the standard deviation.
C. If the graph is skewed, the best method to use for centrality is the mean.
D. If the graph is skewed, the best method to use for centrality is the median.
E. None of the above are true.
8. Suppose that a teacher assigns grades using a normal curve as follows: Students that are one standard deviation or
higher from the mean would receive an A. Students that are within one standard deviation above the mean would
receive a B. Students that are within one standard deviation below the mean would receive a C. Students that
are between one and two standard deviations below the mean would receive a D. Students that are two standard
deviations below the mean would receive an F. The teacher has 200 students. How many of each letter grade
would she expect to give?
The Normal Curve is shown below, with the percentages that correspond to each region. Notice that there are
eight total subregions of the graph.
A-students: These students are one standard deviation or higher, so it’s the three regions on the right. Thus,
13.5% + 2.35% + 0.15% = 16% of the students will receive an A. That’s 200 × 0.16 = 32 students.
B-Students: These students are within one standard deviation above the mean. Thus, it’s the region between µ
an +σ, which is 34%. That’s 200 × 0.34 = 68 students.
C-Students: These students are within one standard deviation below the mean, thus it’s the region that mirror
images the B students on the other side. Thus there will be 68 students that will receive a C.
D-Students: These students are between one and two standard deviations below the mean, so it’s the 13.5%
region between −2σ and −σ. That’s 200 × 0.135 = 27 students.
F-Students: These students fall two standard deviations (or more) below the mean, so these will be the last two
regions on the left. That’s 200 × (0.015 + 0.0235) = 5 students.
9. Heights of female humans are relatively normal with mean µ = 63.7 inches and standard deviation σ = 2.7 inches.
Answer the following.
(a) Cindy falls within one standard deviation of the mean. What range of heights would Cindy fall in?
One Standard Deviation would be [µ − σ, µ + σ] = [63.7 − 2.7, 63.7 + 2.7] = [61.0, 66.4].
(b) Elizabeth considers herself extremely tall (right outlier) at 6’1” (74 inches). Explain (with key statistical
concepts) whether or not she has a valid reason to do so.
To be a right outlier, she needs to be beyond three standard deviations away from the mean. Here,
µ + 3σ = 63.7 + 3(2.7) = 71.8. Since she is taller than this, she can be considered extremely tall.
(c) Frances considers herself extremely short (left outlier) at 4’8” (56 inches). Explain (with key statistical concepts)
whether or not she has a valid reason to do so.
To be a left outlier, she needs to be beyond three standard deviations away from the mean. Here,
µ−3σ = 63.7−3(2.7) = 55.6. Since she is (barely) taller than this, she cannot be considered extremely short.
(d) What percentage of the female population would fall between the heights of 58.3 inches and 69.1 inches?
Using the fact that we have a normal distribution, we can try to apply the Empirical Formula. Here,
we can see that 58.3 = 63.7 - 2(2.7) and 69.1 = 63.7 + 2(2.7). Thus this range of values correspond
to two standard deviations, so we should expect 95% of the population to fall within these values.