Statistics Final Review Solutions
Empirical Rule
1. Fill in the following template using the Empirical Rule and label each segment with the percentage of the whole curve that it represents.
2. What’s another name for the Empirical Rule? How do you know how much of the distribution is within 1 standard deviation of the mean? 2 standard deviations? 3 standard deviations?
Within 1 standard deviation: 68% Within 2 standard deviations: 95% Within 3 standard deviations: 99.7%
3. Suppose that the distribution of monthly earnings for a sample of 2000 people who possess a bachelor’s degree is normally distributed with a mean of $2116 and a standard deviation of $455. Answer the following questions based off of the information given above.
a. What percentage of the individuals with a bachelor’s degree earned less than $1661 per month?
Approximately 16%
b. What percentage of the individuals with a bachelor’s degree earned more than $1206 per month?
Approximately 97.5 %
c. What percentage of the individuals with a bachelor’s degree earned between $3,026 and $3,481 per month?
2.35%
d. Approximately how many individuals with a bachelor’s degree earn between $1206 and $3026?
1,900 individuals
e. In between which two monthly salaries does 68% of the data lie? How about for 95%? 99.7%?
68%: 1,661 & 2,571 95%: 1,206 & 3,026 99.7%: 751 & 3,481
Measures of Center
4. The high school basketball scores were 64, 58, 78, 82, 79, 50 and 79. Calculate the mean, median and mode and determine the best measure of center for this data.
Mean: 70 Median: 78 Mode: 79 The median is the best measure of center.
5. Which measure of center best describes the average life span of the animals?
Mean: 62 Median: 35 Mode: 30
The Median is the best measure of center.
Standard Deviation
6. Write the formula for calculating standard deviation. σ=(X-X)2n-1
For the following sets of data, calculate the mean and standard deviation of the data.
7. The data set below gives the prices (in dollars) of cordless phones at an electronics store.
35, 50, 60, 60, 75, 65, 80
Mean: 60.71 Standard Deviation: 15.12
8. The data set below gives the numbers of home runs for the 10 batters who hit the most home runs during the 2005 Major League Baseball regular season.
51, 48, 47, 46, 45, 43, 41, 40, 40, 39
Mean: 44 Standard Deviation: 4.03
Normal Calculations (Standardized Scores)
9. 600 students took a coordination test. Scores were normally distributed with a mean of 110 and a standard deviation of 16.
a. What percentage of students scored 100 or below? Round to the nearest percent.
27% (if you truncate the z-score of -0.625 to -0.62)
b. What percentage of students scored 100 or above? Round to the nearest percent.
73%
c. What percentage of students scored between 90 and 125? Round to the nearest percent.
72%
d. What percentile would a student rank if they scored an 80?
31st percentile (if you truncate z-score of -1.875 to -1.87)
10. A fifth grader takes a standardized achievement test (mean = 125, standard deviation = 15) and scores a 148. What is the child’s percentile rank?
94th percentile
11. A patient recently diagnosed with Alzheimer’s disease takes a cognitive abilities test and scores a 45. The mean on this test is 52 and the standard deviation is 5. What is the patient’s percentile rank?
8th percentile
12. Three students take equivalent stress tests. Which is the highest relative score (meaning which has the largest z score value)? C has the highest relative score.
a. A score of 144 on a test with a mean of 128 and a standard deviation of 34. 0.47
b. A score of 90 on a test with a mean of 86 and a standard deviation of 18. 0.22
c. A score of 18 on a test with a mean of 15 and a standard deviation of 5. 0.6
13. Ms. Shen gave a test to her Math 1 class and her Advanced Algebra class. The Math 1 test was 100 pts and was normally distributed with a mean of 73 and standard deviation of 5. The Advanced Algebra test was 150 points and was normally distributed with a mean of 110 and standard deviation of 15. If a student in the Math 1 class scored an 80 and a student in the Advanced Algebra class scored a 115, find their standardized scores and determine who scored higher relative to their testing group.
The Math 1 student has the higher relative score.
Math 1 student’s standardized score (z-score): 1.4 Adv. Algebra student’s standardized score: 0.33
Shape
14. Determine the shape of the following graphs.
a. b. c.
Normally distributed Right- Skewed Left-Skewed
15. What is the shape of the data set { 2, 20, 25, 28, 30, 38} ?
Left-Skewed (because of the unusually low value)
Correlation
16. Estimate the correlation coefficient for the following distribution and evaluate the strength and direction of the correlation. R=-0.6, Moderately strong/negative correlation
17. Match the following graphs to the correlation coefficient scale below and then describe the strength and direction of the correlation (for example: r= 0.8; Strong/Positive Correlation).
a. b. c. d.
r=-0.8 moderately strong/ r=0 Weak correlation r= 0.5 Weak/Moderately strong r= 1 Strong/
Negative correlation Positive correlation Positive correlation
e. f. g.
r= 0.8 moderately strong/ r=-0.5 weak/moderately strong r= -1
positive correlation negative correlation Strong Negative Correlation
g. a. f. b. c. e. d.