TC3 → Stan Brown → Statistics → Fall08 ME50 → Chapter 3 Quiz
revised Sep 30, 2008

Quiz with Solutions: Chapter 3 (35 min)

These solutions show about the same level of work I expect from you, though I add quite a bit of extra commentary. Please see Show Your Work for the what, why, and how.

normal curve showing shaded area 1(points: 1.5)  Adult male hobbits’ heights are normally distributed (bell curve) with μ = 36 in and σ = 3 in. About what percent of adult male hobbits are 30 to 42 in tall?

Solution: Since you’re told it’s a normal distribution, you can use the Empirical Rule for normal distributions. Begin by computing the z scores:

z1 = (x1−μ)/σ = (30−36)/3 = −2

z2 = (x2−μ)/σ = (42−36)/3 = +2

The Empirical rule tells you that 95% of the data are within two standard deviations above and below the mean for a normal distribution. Answer: 95%

2(points: 1) Mensa is a society for people with IQs in the top 2% of the population. On an IQ test, what percentile is the threshold to qualify you for Mensa?

Solution: The percentile is the percent of the sample or population with lower scores. 100−2 = 98% of the population has a lower score than the top 2%. Answer: 98th percentile

3(points: 2.5) You’re looking at houses in Abercrombie and Bucks counties. House A in Abercrombie costs $200,000, where the mean house price is $150,000 and the standard deviation is $20,000. House B in Bucks costs $225,000, where the mean is $180,000 and the standard deviation is $25,000. Which house is more expensve relative to the county? Explain.

Solution: z equals (x-bar minus mu) over sigma To compare apples and oranges, compute their z scores:

zA = (200,000−150,000)/20,000 = 50/20 = 2.5

zB = (225,000−180,000)/25,000 = 45/25 = 1.8

Because it has the higher z score, house A is relatively more expensive.

4(points: 1) In a particular data set, the mean is 11.5 and the median is 8.7. What if anything can you say about the shape of the distribution?

Answer: The mean is greater than the median. This usually means that the distribution is skewed right, like incomes at a corporation.

Remark: Though we don’t study them in this class, it’s actually possible to compute numbers that measure the shape of a distribution. If you’re interested, see Skewness and Kurtosis on the TI-83/84.

 

5(points: 6) Here are 19 exam scores taken at random from a class of 100 students:

52 62 66 68 72 74 74 76 76 76 78 78 82 82 84 86 88 92 96

Compute the following quantities and label each with its name and (when one exists) its proper symbol: mode, range, median, mean, standard deviation, IQR, variance. Use any valid method, but show your work.

Solution: mode = 76 because 76 occurs three times and the other numbers occur less often.

range = high−low = 96−52 = 44; range R = 44

For the remaining statistics, enter the numbers in L1 and execute the command 1-VarStats L1. Then you can read off most of the statistics:

median M or x̃ = 76

mean x̄ = 76.9 (rounded from 76.94736842)

standard deviation s = 10.5 (rounded from 10.50647252)

IQR = Q3 − Q1 = 84−72 = 12. IQR = 12

variance = (standard deviation)². On calculator, find Sx² = 110.3859649 and round to s² = 110.4

Common mistake: Because the data set is a sample, not the whole population, the mean and standard deviation are x̄ and s, not μ and σ.

Common mistake: The standard deviation is 10.5 (Sx), not 10.2 (σ), because this is a sample and not the population.

Common mistake: If you square the rounded number 10.5, you get 110.25, which is incorrect for the variance. Remember The Big No-no.

Age of
Mother		 Class
Midpoint		 Freq. of
Mult. Births
15–1917.593
20–2422.5511
25–2927.51628
30–3432.52832
35–3937.51843
40–4442.5377
  Total7284

6(points: 3) The table represents the number of live multiple-delivery births (≥3 babies) in the U.S. in 2002 for women aged 15 to 44. Fill in the middle column, compute the approximate mean and standard deviation of mother’s age, and label them with their names and proper symbols.

Solution: To compute the statistics of a frequency distribution, begin with the class midpoints (also known as class marks), as shown in the table.

Common mistake: The class midpoint is the midpoint between successive lower boundaries, such as ½(15+20) = 17.5. You can’t use the lower and upper bounds of a class and compute ½(15+19) = 17. See page 145 in your text.

Enter the class midpoints in L1 and the frequencies in L2. Execute 1-VarStats L1,L2 and verify that n = 7284 in the output. Read off the desired statistics:

μ = 32.3 (rounded from x̄ = 32.27210324)

σ = 5.2 (rounded from σx = 5.188930412)

Common mistake: If you use 1-VarStats L1 instead of the correct command, you’ll have n = 6 and wrong statistics. Always check n in the output to guard against gross errors like this.

Common mistake: This data set is the whole population, so you must use μ not x̄, σ not s.

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