# Mathematical statistics

1. For the following order of the measures of centrality, draw the shape that best describes the associated distribution.

2. In a data set with 25 scores, if the value 16 occurs 12 times, what is its relative frequency?

3. Find the median of the following data set.

8, 7, 2, 4, 4, 8, 2, 4, 6, 5, 7, 3, 9, 4, 5, 4, 7

4. List the corresponding deviations for the following set of raw scores:
7, 8, 10, 10, 11, 12, 13, 13

5. Five students took a personality test. Their deviation scores were –4, +3, +1, –2, and +2. Assuming that these scores form a normal distribution, which deviation represents the highest raw score?

6. If a distribution has 45% of scores between the mean of 75 and the score of 30, and 45% of scores between the mean of 75 and the score of 120, what is the total percentage of the distribution in the tails?

7. In a normal distribution if 5% of the scores are greater than 80, what percentage of the scores are between the mean and 80?

8. The Red Cross wanted to know whether an ad campaign designed to persuade people to donate blood actually increased blood donations. For a sample of N = 4 blood donation centers, the mean increase in blood donations was 10 pints, with a standard deviation of 2 pints. What would you estimate the population standard deviation to be?

9. Ms. Gonzalez has recorded the test scores for each of her 25 students. She has calculated = 320 and = 5,000. Calculate the estimated population standard deviation for her data.

10. Frankie has data from a sample of 45 people analyzing their knowledge about cholesterol. He has calculated = 600 and = 90. Calculate the sample standard deviation for these data.

11. A group of 10 students participated in a study of nutritional awareness. What is the sample variance of their scores?

20, 19, 22, 20, 15, 18, 15, 21, 19, 20

12. A group of 10 students participated in a study of nutritional awareness. What is the estimated population standard deviation ?

20, 19, 22, 20, 15, 18, 15, 21, 19, 20

13. June has obtained a sample of 150 people from their school records about their career aptitude. She has calculated = 867 and = 38,427. Calculate the sample standard deviation for these data.

14. In a normal distribution if 35% of the scores are between the score of 40 and the mean of 60, what percentage of the scores is below 40?

15. If there were 500 students in Jamal’s class, approximately how many students scored higher than Jamal on the quiz if Jamal had a z-score of 1.21?

16. Jason’s z-score for his time in the 10-K race was 3.3. If the raw score standard deviation was 5 and the mean running time for the competitors was 55 minutes, what was his raw score?

17. If only the upper 30.15% (a proportion of .3015) of a normally distributed class passed a quiz for which the mean was 70 and the standard deviation was 10, what was the approximate lowest passing score? Show your work.
18. The sample mean for a recent introductory psychology test was 78 and the sample variance was 9. Approximately what proportion of the scores was above 75?

19. The body mass index (BMI) was determined for 500 sixty-year-old males (N=500) and resulted in a perfectly normal distribution. The mean BMI score was 29 with a standard deviation of 6. Answer the following questions:

a. Approximately how many males have a BMI score less than 17?

b. What is the range of BMI scores for the middle 68% chunk of the distribution?

c. What is the probability that a 60-year-old male has a BMI less than 35?

d. What is the median BMI score?

e. What is the probability that a 60-year-old male has BMI of 38.87 or greater?

20. Suppose that on average, students are absent 38 days per year, with a standard deviation of 5 days. (3 points)
A. What percentage of students will miss 41 or fewer days in a year?

B. What percentage of students will miss more than 49 or more days of school?

C. What percentage of students will miss 38 or fewer days?

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