Download 10] Descriptive Statistics Measures

10] Descriptive Statistics Measures
10.1) Describe what statistics is about.
10.2) Measures of Central Tendency: Mean, Median, Mode, Midrange
a) Find the mean of the sample of scores: 20, 13, 29, 8, 18, 24, 6.
b) The weights (in pounds) of 12 newborn babies were recorded as
{7.9, 6.8, 7.4, 7.5, 7.6, 7.5, 7.6, 6.9, 7.9, 7.5, 6.5, 7.1}. Find the
mean weight of these newborn babies. ANSWER: x =
= 7.35 pounds
c) Case 1: Find the median of the sample 65, 83, 45, 52, 41, 33, 98
Case 2: Find the median of: 73, 81, 44, 51, 41, 34, 90, 62
d) Find the mode(s) in each sample: d1) 3, 7, 2, 7, 1, 6, 7, 6, 3;
d2) 70, 61, 41, 52, 61, 52; d3) 30, 35, 30, 50, 45, 30, 45, 50, 45, 35, 35
88.2
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d4) 3, 3, 3, 3, 7, 7, 7, 7, 21, 21, 21, 21, 9, 9, 9, 9, 1, 1, 1, 1, 6, 4, 4, 4
d5) 30, 42, 58, 71, 90, 94, 99; d6) 4, 4, 4, 4, 5, 5, 5, 5, 8, 8, 8, 8
e) Find the midrange for the samples in parts a) and b).
10.3) Measures of Variation: Range, Variance, Standard Deviation.
10.4) Use the sample of numerical scores {12, −4, 12, 7, 12, 7} to find
the: a) mean; b) median; c) mode; d) midrange; e) range; f) variance
(use different computational formulas); g) standard deviation.
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∑ x = 46; ∑ ( x − x ) = 38.667; ∑ x = 546; a) 7.667; b) 9.5; c) 12; d) 4; e) 8; f) 38.667; g) 6.218
10.5) Use the sample given in 10.2)a) to find the: a) mean; b) median; c) mode;
d) midrange; e) range; f) variance (use different computational formulas);
g) standard deviation. Answers: (a) 16.857; b) 18; c) no mode; d) 17.5; e) 23; f) 70.143; g) 8.375
10.6) Use the sample given in 10.2)b) to find the: a) mean; b) median; c) mode;
d) midrange; e) range; f) variance (use different computational formulas);
g) standard deviation. Answers: (a) 7.308; b) 7.45; c) 7.5; d) 7.2; e) 1.4; f) 0.161; g) 0.401
10.7) Mean, Median, and Mode in Skewed Distributions
Skewed to the left
Symmetric
Skewed to the right
10.8) Empirical Rule
10.9) Z-Scores (or Standardized Scores)
10.10) A distribution has mean 75 and standard deviation 6. Compute
and interpret the z-score for the numbers: (a) 85; (b) 63; (c) What
number corresponds to z = −2.67 ? (d) Find the value of the number
whose z-score is 0.54. ANSWERS: (a) 1.67; (b) −2; (c) 58.98; (d) 78.24
10.11) A perfectly symmetric distribution has mean 250 and standard
deviation 12. Use the empirical rule to find the approximate percent
of scores described as follows: a) less than 262; b) greater than 274;
c) between 226 and 286; d) at least 226; e) at most 274; f) no less than
214 and no more than 250; g) do not exceed 286; h) minimum 214.
ANSWERS: a) ~84%; b) ~2,5%; c) ~97%; d) ~97.5%; e) ~97.5%; f) ~49.5%; g) ~99.5%; h) ~99.5%
10.12) A perfectly symmetric distribution has mean 83 and standard deviation 6. Based on
the empirical rule, between what values can we expect to find the approximately:
a) 68% of the scores? b) 95% of the scores? c) 99.7% of the scores?
ANSWERS: a) between 77 and 89; b) between 71 and 95; c) between 65 and 101
10.13) Answer true or false in each case (assume a perfectly symmetric distribution):
a) the percent of scores one sd above the mean is greater than the percent of scores one
sd below the mean; b) the percent of scores between 2 sd and 1 sd below the mean is
less than the percent of scores between 1 sd and 2 sd above the mean; c) the percent of
scores between 3 sd and 2 sd below the mean is equal to the percent of scores between 2
sd and 3 sd above the mean; d) the percent of scores between the mean and 1 sd below
the mean is equal to the percent of scores between 1 sd and 2 sd above the mean; e) the
percent of scores between 1 sd and 2 sd above the mean is equal to the percent of scores
between 2 sd and 3 sd above the mean; f) the percent of scores between 2 sd and 1 sd
below the mean is more than the percent of scores between 3 sd and 2 sd below the
mean; g) the median is less than the mean; h) the mean is greater than the mode; i) the
median and the mode are equal; j) the mean and the median are equal.
ANSWERS: a) false; b) false; c) true; d) false; e) false; f) true; g) false; h) false; i) true; j) true
10.14) Answer true or false in each case (relationship between mean, median, mode in
skewed distributions): a) In positively skewed distributions, the mean is the largest of
the three values; b) in negatively skewed distributions, the mean is less than the median;
c) in positively skewed distributions, the median is the greater than the mean; d) in
negatively skewed distributions, the median is less than the mode; e) in positively
skewed distributions, the median is equal to the mode; f) in negatively skewed
distributions, the mean is greater than the median. Ans: true; true; false; true; false; false
10.15) A symmetric distribution has mean 109 and standard deviation 25; a) Compute its
variance; b) Find and interpret the z-score for the number 165; c) Find the number with
z = −1.35; d) Use the empirical rule to approximate the percent of values that are less
than 84; e) Answer true or false: the percent of scores between 34 and 59 is the same as
the percent of scores between 159 and 184. Answers: a) 625; b) 2.24; c) 75.25; d) ~16%; e) true
10.16) Suppose that 79.75 and 95.90 are observations from a population of scores and
they have z = −1.4 and z = 2, respectively. Find the mean and the standard deviation
of the population.
Answers: µ = 86.4; σ = 4.75
10.17) A web site for prospective students at a particular university states that full-time
students spend an average of $325 on text-books each semester. The standard
deviation of the amount that full-time students spend on text-books each semester is
known to be $52; a) If the $325 value is correct and the distribution is symmetric, do
many students spend less than $221 on textbooks each semester? Explain; b) If the
$325 value is correct and the distribution is symmetric, what are the least and the most
amounts that virtually all students spend per semester on text-books? c) While buying
text-books at the bookstore, you notice that the student making the line ahead of you
spends $550 for text-books that semester. Does this make you suspect that perhaps the
$325 advertised by the university is incorrect? Explain.
Ans: a) no, only approximately 2.5%; b) $169 to $481; c) yes, $550 is more than 3 sd (4.33 sd) above the mean
10.18) A manufacturer of fluorescent light-bulbs claims that the bulbs she produces have
a mean lifetime of 1450 hours. Assume the distribution of lifetime values is mound
shaped with standard deviation 50 hours; a) If the manufacturer’s claim is true,
approximately what percent of light-bulbs will last more than 1500 hours? b) If the
manufacturer’s claim is true, approximately what percent of light-bulbs will last more
than 1350 hours? c) Suppose you purchase a fluorescent light-bulb from this
manufacturer. If it lasts 1250 hours, could you infer that perhaps the manufacturer’s
claim is false? Explain.
Answers: a) ~16%; b) ~97.5%; c) yes, 1250 is more than 3 sd (4 sd) below the mean