Illinois State University Mathematics Department
MAT 312: Probability and Statistics for Middle School Teachers
Test #1: Possible Solutions
- Part I: 25 Multiple Choice Questions (1 pt each)
- Part II: 2 Open-Response Questions (10,15 pts)
- Total: 50 points
- Impact on Course Grade: 20% of your Course Grade
Criteria Used to Evaluate Part II Responses
26: 10 points (2 pts each)
- 1 pts for each numerical response
- 1 pt for clear and accurate explanation
27: 15 points
- 10 pts for appropriate and accurate evidence
- 5 pts for clear and accurate explanation
Part I: Multiple Choice
For each question, choose the one best response and circle that letter at the appropriate spot on the answer sheet.
1. Which visual representation does not preserve the values in a data set?
- a. box-and-whiskers plot
- b. line plot
- c. 5-number summary
- d. stem-and-leaf plot
- e. More than one of the visual representations do not preserve the values in a data set.
- f. None of the visual representations preserve the values in a data set.
For items 2-5, select one of the following data types to best describe each variable.
a. interval data
b. ordinal data
c. ratio data
d. nominal data
2. length of flights, in hours and minutes, for a commerical airline
3. hair color of flight attendants of a commerical airline
4. flight departure times for a commercial airline
5. number of passengers flown by a commercial airline
For questions 6-9, use the following data set: 3,4,7,8,9,9,12,17,17,17,18
6. Determine the lowspread of the data.
- a. 4
- b. 6
- c. 8
- d. 10
7. Determine the mean of the data.
- a. 7
- b. 9
- c. 11
- d. 13
8. Determine the upper inner fence of the data.
- a. 15
- b. 17
- c. 24
- d. 32
9. Determine the sum of squares of the data.
- a. 0
- b. 5.26
- c. 27.64
- d. 304
10. Consider the following data set: 13, 18, 28, 28, 31, 23, 35, 19. Which measure of central tendency will change the most if the "23" should have been a "32"?
- a. median
- b. mode
- c. mean
- d. None of these three measures of central tendency will change.
11. What characteristic describes the deviation in a one-variable data set?
- a. strength
- b. shape
- c. spread
- d. location
Use the diagram below for questions 12 through 15.
12. Estimate the location of the 25th percentile.
- a. 8 participants
- b. 15 participants
- c. 32 participants
- d. somewhere between 40 and 50 participants
13. Which quartile in the data set exhibits the least deviation?
- a. the first quartile (0th to 25th percentiles)
- b. the second quartile (25th to 50th percentiles)
- c. the third quartile (50th to 75th percentiles)
- d. the fourth quartile (75th to 100th percentiles)
- e. More than one of the quartiles satisfies the conditions stated.
- f. None of the quartiles satisfy the conditions stated.
14. Estimate the value in the data set that is the largest value inside the upper inner fence.
- a. 8 participants
- b. 32 participants
- c. 80 participants
- d. 94 participants
- e. No such value can be determined.
15. What could be the fewest number of junior golf tournaments represented by this plot?
- a. 3
- b. 8
- c. 12
- d. 16
- e. 32
16. In which situation is it appropriate to use the median as the preferred measure of central tendency?
- a. in reporting average selling price for homes in a community
- b. when the distribution is significantly skewed to the right
- c. in determining what size sweaters to reorder in a retail establishment
- d. when the data is considered bimodal data
17. In a positively skewed distribution, which of the following is most likely true?
- a. The mean and median are approximately equal.
- b. The mean is greater than the median.
- c. The mean is less than the median.
- d. The median takes on the value 0.
18. Suppose that the distribution of the life span of a certain African hornet has a symmetric, mound-shaped (normal) distribution with a mean of 840 hours and a standard deviation of 60 hours. Within what bounds do we expect approximately 68% of the life spans of such hornets to fall?
- a. 720 to 960 hours
- b. 780 to 900 hours
- c. 810 to 870 hours
- d. 840 to 900 hours
For questions 19-22, use the following setting.Thirty (30) couples in a marathon dance club sought to break the record for consecutive hours danced by a couple. Suppose the data shown here represents the consecutive hours danced by each of the 30 couples.
19. Of the 30 couples, what is the relative frequency of couples dancing less than 500 hours?
- a. 10/30
- b. 11/30
- c. 10
- d. 11
20. What value represents the 75th percentile of the observations?
- a. 480 hrs
- b. 555 hrs
- c. 630 hrs
- d. 710 hrs
21. What is the highspread of the dancing data?
- a. 150 hours
- b. 480 hours
- c. 510 hours
- d. 630 hours
- e. None of the values in (a) through (d) represent the highspread.
22. The distribution shown in the plot appears to be _?_.
- a. symmetric
- b. uniform
- c. negatively skewed
- d. positively skewed
23. Which of the following correctly represents the measures of central tendency for the distribution shown here?
- a. X: mean, Y: median, Z: mode
- b. X: mode, Y: mean, Z: median
- c. X: median, Y: mode, Z: mean
- d. X: median, Y: mean, Z: mode
- e. X: mode, Y: median, Z: mean
- f. None of these are correct.
Use the following information for questions 24 and 25.
Sodium has long been a major dietary factor in reducing the risk of, and controlling, high blood pressure.
Under FDA's food labeling rules, the Daily Value for sodium is 2,400 mg. FDA established this value because it is consistent with recommendations and government reports that encourage reduced sodium intakes.
Source: FDA website (www.cfsan.fda.gov/~dms/fdsodium.html) Available 16 June 2004
Its package label claims that one serving (1 ounce) of the Frito-Lay snack food called Cheetos Crunchy contains 290 mg of sodium. Because of measurement and manufacturing variations, suppose that the amount of sodium in any actual serving of Cheetos is normally distributed with a mean of 290 mg and a standard deviation of 16 mg.
24. Based on this information, where can you expect about 95% of the sodium measurements for 1-ounce servings of Cheetos to fall?
- a. between 258 and 322 mg
- b. between 274 and 306 mg
- c. between 282 and 298 mg
- d. at least 290 mg
25. Suppose that Edith, a researcher for the FDA, walked into a meeting and announced that she had just tested a 1-ounce serving of some snack food and determined that it contained 260 mg sodium. Edith did not reveal what specific snack food she had tested. Which of the following justfied conclusions can most likely be drawn about the snack food Edith tested?a. The snack food could not be Cheetos, because the amount of sodium was 30 mg below the mean amount for Cheetos.
b. The snack food could be Cheetos, because the amount of sodium was only 30 mg below the mean amount for Cheetos.
c. The snack food could be Cheetos, because 260 mg is within two standard deviations of the mean amount of sodium in a serving of Cheetos.
d. The snack food could not be Cheetos, because 260 mg is almost two standard deviations less than the mean amount of sodium in a serving of Cheetos.
Part II: Open Response
Respond to each question and write your response in the space provided on the answer sheet.
26. Here is an absolute frequency histogram that shows average mecury levels (expressed in parts per million, or ppm) of fish in 53 lakes in Florida. Use the histogram to answer questions (a) through (e) below. Include a brief note to describe how you arrived at your solution response.a. In what measurement class is the median of the data? Explain how you know.
First, we note that there are five measurement classes and that the sum of the values represented in each measurement class must be 53 lakes. By estimating the height of each rectangle compared to the vertical scale, we see that there are 17 lakes in the first (left-most) measurement class, 18 lakes in the next, 10 lakes in the middle measurement class, 6 lakes in the next, and 2 lakes in the measurement class representing the greatest average mercury level.
The middle data value in a set of 53 ordered values will be the 27th value, so the median of this data lies in the second measurement class, the one representing an average of 0.3 to 0.6 ppm mercury in the lake's fish.
b. In what measurement class does the 75th percentile lie? Explain how you know.
Using the same information described in the first paragraph for the solution to (a) above, we know the 75th percentile will be midway between the 40th and 41st data values, ordered least to greatest. These two values both occur in the measurement class representing an average of 0.6 to 0.9 ppm mercury in the lake's fish.
c. Does this histogram show a symmetric distribution? Explain your response.
The histogram does not show a symmetric distribution, for there is not a line of reflection in the middle of the histogram that would make the left side a reflection of the right. This distribution is positively skewed.
d. Suppose a lake was chosen at random from the 53 lakes whose fish were tested and the average mecury level for fish in that lake was less than 0.5 ppm. What is the largest number of lakes from which this lake could have been chosen? Explain how you know.
This lake had to come from one of the first two (left-most) measurement classes shown above, so this lake could have been chosen from as many as 35 lakes.
e. Suppose that fish containing more than 1.0 ppm mercury is considered potentially harmful for human consumption. A letter is being sent to residents on lakes with average mecury levels of at least 1.0 ppm. What is the largest possible number of lakes whose residents must be sent letters? Explain how you know.
Lakes containing fish with average mercury levels of more than 1.0 ppm are contained in the top two measurement classes. There could be as many as 8 lakes whose residents must be sent letters.
27. The reputation of a steel processing plant rests on its capacity to manufacture products within narrow tolerance limits. Samples of 30 ball bearings advertised by Boston Bearing Company to be 8 mm in diameter were measured and compared to a sample of 30 ball bearings manufactured by Bismarck Bearing Associates, also advertised to be 8 mm in diameter. Here are the data from the two samples.
Boston Bearing Company:
Diameters of 30 bearings (mm)
Bismarck Bearing Associates:
Diameters of 30 bearings (mm)
7.93 7.87 7.84 8.09 8.14 8.28 7.88 8.01 7.92 7.98 8.01 8.05 8.01 8.07 7.94 8.03 8.14 8.09 7.98 8.01 7.93 7.98 8.09 7.95 8.04 8.13 7.90 7.89 8.18 8.13
8.01 8.04 8.09 7.98 8.05 8.10 8.05 8.09 7.99 7.99 7.97 7.99 8.00 8.03 8.03 7.94 7.97 8.03 7.93 8.10 8.00 8.03 8.06 8.08 8.11 8.14 7.96 8.05 8.08 8.00
Analyze these data to determine which of the two companies is the more accurate maufacturer of 8 mm ball bearings. Show all evidence you generated to address the question and write a concise paragraph to summarize your findings.
The 30 ball bearings from Boston have a mean diameter of 8.0163 mm and a sample standard deviation of 0.1038 mm. From Bismarck, the ball bearings have a mean diameter of 8.0296 mm with a sample standard deviation of 0.0535 mm.
Although the mean diameter of the Boston bearings is slightly closer to 8 mm than that of the Bismarck bearings, the standard deviation for the Bismarck bearings is about half that of Boston's. That is, the Bismarck bearings have a smaller deviation from the mean. Because the difference in the mean diameters is so small relative to the difference in the standard deviations, and because most of the diameters will be within one standard deviation of the mean, the bearings from Bismarck have smaller deviations from 8 mm.