In an editorial, the Poughkeepsie Journal printed this statement: "The median price – the price exactly in between the highest and lowest – ..." Does this statement correctly describe the median? Why or why not?
No. It describes the midrange, not the median.
The measure of center that is the value that occurs with the greatest frequency is the _______.
mode
A value at the center or middle of a data set is a(n) _______.
measure of center
Which of the following is always true?
A. Data skewed to the right have a longer left tail than right tail.
B. In a symmetric and bell-shaped distribution, the mean, median, and mode are the same.
C. For skewed data, the mode is farther out in the longer tail than the median.
D. The mean and median should be used to identify the shape of the distribution.
B. In a symmetric and bell-shaped distribution, the mean, median, and mode are the same.
Pennies made before 1983 are 97% copper and 3% zinc, whereas pennies made after 1983 are 3% copper and 97% zinc. Listed below are the weights (in grams) of pennies from each of the two time periods.
Find the mean and median for each of the two samples.
Does there appear to be a considerable difference in the means?
The mean weight of the pennies made before 1983 is 3.1133 grams.
The median weight of the pennies made before 1983 is 3.10645 grams.
The mean weight of the pennies made after 1983 is 2.4926 grams.
The median weight of the pennies made after 1983 is 2.49315 grams.
Yes, because the difference in the means is more than 5%.
Listed below are head injury measurements from small cars that were tested in crashes. The measurements are in "hic," which is a measurement of a standard "head injury criterion," (lower "hic" values correspond to safer cars). The listed values correspond to cars A, B, C, D, E, F, and G, respectively.
Find the a. mean, b. median, c. midrange, and d. mode for the data.
e. Which car appears to be the safest?
f. Based on these limited results, do small cars appear to have about the same risk of head injury in a crash?
a. The mean is 417.7.
b. The median is 392.
c. The midrange is 425.
d. There is no mode.
e. car E
f. No, because the data values differ substantially.
Because the mean is very sensitive to extreme values, it is not a resistant measure of center. The trimmed mean is more resistant. To find the 10% trimmed mean for a data set, first arrange the data in order, then delete the bottom 10% of the values and the top 10% of the values, then calculate the mean of the remaining values.
For the following credit-rating scores, find (a) the mean, (b) the 10% trimmed mean, and (c) the 20% trimmed mean.
How do the results compare?
a. The mean is 720.5.
b. The 10% trimmed mean is 729.3.
c. The 20% trimmed mean is 734.3.
The distribution of the data appears to be skewed to the left because the results appear to show a trend of increasing values as the percentage of trim increases.
Listed below are the durations (in hours) of a simple random sample of all flights of a space shuttle.
Find the (a) mean, (b) median, (c) mode, and (d) midrange for the given sample data.
(e) Is there a duration time that is very unusual? How might that duration time be explained?
a. The mean is 218.7 hours.
b. The median is 239.0 hours.
c. The mode is 239 hours.
d. The midrange is 192.0 hours.
e. Yes, the time of 0 hours is very unusual. It could represent a flight that was aborted.
An experiment was conducted to determine whether a deficiency of carbon dioxide in the soil affects the phenotype of peas. Listed below are the phenotype codes where 1 equals smooth dash yellow, 2 equals smooth dash green, 3 equals wrinkled dash yellow, and 4 equals wrinkled dash green.
Find the (a) mean, (b) median, (c) mode, and (d) midrange for the given sample data.
(e) Do the results make sense?
a. The mean phenotype code is 2.9.
b. The median phenotype code is 3.0.
c. The mode phenotype code is 3.
d. The midrange of the phenotype codes is 2.5.
e. Only the mode makes sense since the data is nominal.
Refer to the data set of times, in minutes, required for an airplane to taxi out for takeoff, listed below.
Find the mean and median.
How is it helpful to find the mean?
The mean of the data set is 28.5 minutes.
The median of the data set is 27.0 minutes.
The mean taxi out time is important for calculating and scheduling the arrival time.
Below are the range and standard deviation for a set of data.
Use the range rule of thumb and compare it to the standard deviation listed below.
Does the range rule of thumb produce an acceptable approximation? Suppose a researcher deems the approximation as acceptable if it has an error less than 15%.
The estimated standard deviation is 0.730
No, because the error of the range rule of thumb's approximation is greater than 15%.
Cans of regular soda have volumes with a mean of 12.01 oz and a standard deviation of 0.11 oz.
Is it unusual for a can to contain 12.11 oz of soda?
Minimum "usual" value = 11.79 oz
minimum "usual" value = (mean) – 2 x (standard deviation
Maximum "usual" value = 12.23 oz
minimum "usual" value = (mean) + 2 x (standard deviation)
No, because it is between the minimum and maxmum "usual" values.
Cans of regular soda have volumes with a mean of 12.24 oz and a standard deviation of 0.11 oz.
Is it unusual for a can to contain 12.58 oz of soda?
Minimum "usual" value = 12.02 oz
minimum "usual" value = (mean) – 2 x (standard deviation
Maximum "usual" value = 12.46 oz
minimum "usual" value = (mean) + 2 x (standard deviation)
Yes, because it is larger than the maximum "usual" value.
Identify the symbols used for each of the following: (a) sample standard deviation; (b) population standard deviation; (c) sample variance; (d) population variance.
a. The symbol for sample standard deviation is s.
b. The symbol for population standard deviation is σ .
c. The symbol for sample variance is s²
d. The symbol for population variance is σ² .
The trend of thinner beauty pageant winners has generated charges that the contest encourages unhealthy diet habits among young women. Listed below are body mass indexes (BMI) for beauty pageant winners from two different time periods.
Find the coefficient of variation for each of the two sets of data.
Is there a difference in variation between the two data sets?
The coefficient of variation for the BMI's of beauty pageant winners from the 1920s and 1930s is 7.50%.
CV = stand dev ÷ mean x 100%
The coefficient of variation for the BMI's of recent beauty pageant winners is 6.37%.
There is no significant difference in the variations.
Waiting times (in minutes) of customers at a bank where all customers enter a single waiting line and a bank where customers wait in individual lines at three different teller windows are listed below.
Find the coefficient of variation for each of the two sets of data.
Is there a difference in variation between the two data sets?
The coefficient of variation for the waiting times at Bank A is 6.41%.
CV = stand dev ÷ mean x 100%
The coefficient of variation for the waiting times at the Bank B is 24.61%.
The waiting times at Bank A have considerably less variation than the waiting times at Bank B.
Which of the following is NOT a property of the standard deviation?
A. When comparing variation in samples with very different means, it is good practice to compare the two sample standard deviations.
B. The value of the standard deviation is never negative.
C. The standard deviation is a measure of variation of all data values from the mean.
D. The units of the standard deviation are the same as the units of the original data.
A. When comparing variation in samples with very different means, it is good practice to compare the two sample standard deviations.
The square of the standard deviation is called the _______.
variance
Listed below are the gross amounts (in millions of dollars) earned in box office receipts for a recent movie. The amounts are listed in order for the first 14 days of the movies release.
Find the range, variance, and standard deviation of the data set.
If you invested in this movie, what characteristic of the data set would you care about most, and is it a measure of center or variation?
The range of the sample data is 55 million dollars.
The variance of the sample data is 366.8 million dollars².
The standard deviation of the sample data is 19.2 million dollars.
The gross from opening day and the rate of decline. The measures of center and variation are less important.
Listed below are the amounts of mercury (in parts per million, or ppm) found in tuna sushi sampled at different stores.
Find the range, variance, and standard deviation for the set of data.
What would be the values of the measures of variation if the tuna sushi contained no mercury?
The range of the sample data is 0.48 ppm.
Sample variance = 0.029 ppm²
Sample standard deviation = 0.171 ppm
The measures of variation would all be 0.
Heights of men on a baseball team have a bell-shaped distribution with a mean of 182 cm and a standard deviation of 7 cm.
Using the empirical rule, what is the approximate percentage of the men between the following values?
a. 168 cm and 196 cm
b. 161 cm and 203 cm
a. 95% of the men are between 168 cm and 196 cm.
b. 99.7% of the men are between 161 cm and 203 cm.
- one standard deviation from the mean accounts for about 68% of the set
- two standard deviations from the mean account for about 95%
- three standard deviations from the mean account for about 99.7%.
The following are amounts of time (minutes) spent on hygiene and grooming in the morning by survey respondents.
Determine the 5-number summary and construct a boxplot for the data given below.
The 5-number summary is 6, 11, 18.5, 37, 62.
Below are 36 sorted ages of an acting award winner.
Find P_{70} using the method presented in the textbook.
P_{70} = 66
L = k ÷ 100 x n
70 ÷ 100 x 36 = 25.2 (round up)
What is the Lth = 26th value in the sorted list? 66
What is the next value in the sorted list? 66
P_{75} = (66 + 66) ÷ 2 = 66
IQ scores are measured with a test designed so that the mean is 103 and the standard deviation is 13. Consider the group of IQ scores that are unusual.
What are the z scores that separate the unusual IQ scores from those that are usual?
What are the IQ scores that separate the unusual IQ scores from those that are usual?(Consider a value to be unusual if its z score is less than –2 or greater than 2.)
The lower z score boundary is –2.
The higher z score boundary is 2.
The lower bound IQ score is 64.
(lower z score x standard deviation) + mean
The higher bound IQ score is 136.
(higher z score x standard deviation) + mean
If your score on your next statistics test is converted to a z score, which of these z scores would you prefer: –2.00, –1.00, 0, 1.00, 2.00? Why?
The z score of 2.00 is most preferable because it is 2.00 standard deviations above the mean and would correspond to the highest of the five different possible test scores.
Use the same scale to construct boxplots for the ages of the best actors and best actresses from the accompanying data sets.
Use the boxplots to compare the two data sets.
Actors: 27, 37, 43, 50, 76
Actresses: 21, 28, 33, 39, 82
Although actresses include the oldest age, the boxplot representing actresses shows that they have ages that are generally lower than those of actors.
Whenever a data value is less than the mean, _______.
the corresponding z-score is negative
Which is relatively better: a score of 76 on a psychology test or a score of 45 on an economics test? Scores on the psychology test have a mean of 89 and a standard deviation of 5. Scores on the economics test have a mean of 55 and a standard deviation of 7.
The economics test score is relatively better because its z score is greater than the z score for the psychology test score.
Psych = (76 – 89) ÷ 5 = –2.6
Econ = (45 – 55) ÷ 7 = –1.428571429
Below are 36 sorted ages of an acting award winner.
Find P_{25} using the method presented in the textbook.
P_{25} = 29.5
L = k ÷ 100 x n
25 ÷ 100 x 36 = 9
What is the Lth = 9th value in the sorted list? 29
What is the next value in the sorted list? 30
P_{25} = (29 + 30) ÷ 2 = 29.5