Module 5

A college statistics class conducted a survey of how students spend their money. They asked 25 students to estimate how much money they typically spend each week on fast food. They determined that the mean amount spent on fast food each week is $31.52 and the median is $32. Later they realized that a value entered as $2 should have been $20. They recalculate the mean and the median. Which of the following is true? 1. The mean and median will increase. 2. The mean will increase, but the median will remain the same. 3. The mean will stay the same, but the median will increase. 4. Both the mean and median will remain the same.

2 The mean increases because we are adding $20 instead of $2 to the overall sum. The median does not change because $20 is still in the lower half of the data. So the position of the middle score remains the same

What is your estimate for the measure of center you chose?

36 36 cm is a good estimate for the median. Approximately 50% of the adults have calf measurements above 36 cm and approximately 50% of the adults have calf measurements below 36 cm.

This histogram shows the distribution of exam scores for a class of 33 students. Which of the following is the most reasonable estimate for the median? 1. 50 2. 75 3. 85

75 There are 33 students, so the median has 16 scores above it and 16 scores below it (33/2 = 16.5). So the median falls between 70 and 80.

At a large community college, full-time students enroll in a minimum of 12 credit hours and a maximum of 18 credit hours each semester. Students must petition for approval to enroll in more than 18 credit hours. The histogram shows the distribution of credit hours for 31 randomly selected full-time students. Which is the most appropriate description of how to determine a typical number of credit hours for full-time students? 1. The median is halfway between the smallest and the largest number of credit hours, so the median is 17. Since the bulk of the data is to the left of the median, the mean is pulled left. The mean is about 16. X 2. Students must petition for approval to enroll in more than 18 hours, so any values above 18 are outliers and should be disregarded. This makes the data approximately symmetrical with a central peak. Therefore the mean and the median are both about 15, and both are appropriate representations of the typical number of credit hours. 3. The median is 15. The distribution is skewed right, so the mean will be greater than the median. For this reason, the median is a better representation of the typical number of credit hours.

Answer: 3 There are 31 students, so there are 31 credit-hour data points. There will be 15 data points below the median and 15 above (31/2 = 15.5). There are 10 students in the first three bins, and the 15th and 16th students are in the 5th bin. The 5th bin represents 15 credit hours, so the median number of credit hours is 15. The data is skewed right, so the mean will be pulled to the right. So the median is the best representation of the typical number of credit hours.

The histogram shows the distribution of the scores on a statistics exam in a large class of 70 students. Which statement below is most accurate in describing the center of the distribution? 1. The median should be used to describe the center, because it is halfway between the largest and smallest exam scores. 2. The mean should be used to describe the center because the data is not symmetric. 3. The mean should be used because it may produce a decimal value which is more accurate. 4. The median should be used to describe the center. The data is skewed, so the middle outcome more accurately describes the typical exam score.

Answer: 4 Yes, we should not use the mean to measure the center of skewed data sets and instead use the median as a more accurate measure of center.

What is the mean of the dotplot pictured? 1. 3 2. 4 3. 5 4. 4.44

Answer: 4 First we add the data values to get 0+1+3+3+3+4+5+6+6+9 = 40. Then we divide by the number of data values 40 ÷ 10 = 4.

The school committee of a small town wants to determine the average number of children per household in their town. There are 50 households in the town. They divide the total number of children in the town by 50 and determine that the average number of children per household is 2.2. Which of the following must be true? 1. The most common number of children in a household is 2.2. 2. None of the other statements is true. 3. Half of the households in the town have more than 2 children. 4. There are a total of 110 children in the town.

Answer: 4. There a total of 110 children in the town. Correct. The mean is the "fair share" measure of center. So we can think of 50 households each with 2.2 children. There are 2.2 x 50 = 110 total children. (Obviously, a single household cannot really have 2.2 children.)

Jose wants to have an average of 80 for his 4 exams. Each exam is scored on a scale of 0 to 100. His first three exam scores are: 87, 72, 85. What does Jose need to score on the 4th exam to have a mean of 80 on all 4 exams?

Answer: 76 The mean is the "fair share" measure of center. We can think of the mean as the score he would make on all four exams if he scored the same on every exam. If he scored an 80 on every exam, he has 80 x 4 = 320 points. He has 87 + 72 + 85 = 244 points now. So he needs 320-244 = 76 points on the 4th exam. To check our work, let's find the mean. Mean = (87 + 72 + 85 + 76) / 4= 320 / 4 = 80

Which distribution has the smallest mean? 1. Set A 2. Set B 3. Set C

Answer: Set A Set A has the smallest mean at about 72. Typical values are at 70 and 71, but the right skew pulls the mean toward the outliers.

For the 17 quiz scores graphed in the histogram, estimate the median. 1. between 16 and 18 2. between 10 and 12 3. 13

Between 16 and 18 There are 17 scores, the median will have 8 scores below it and 8 scores above it. There are 8 scores between 4 and 16. So the median could be 16 or any number between 16 and 18.

The dotplot above shows the distribution of calf measurements for 507 adults who attend a local gym. The data is divided into equal-sized subgroups with 25% of the adults in each subgroup. Which measure of center can you most accurately estimate from this information? 1. mean 2. median

Median The median divides the data into equal-sized groups. So approximately 50% of the data is above the median and approximately 50% of the data is below the median.

Which measure of center is the best measure of a typical value for the data set in the histogram above? 1. Median 2. Mean 3. Both

This data set is fairly symmetric with no outliers, so both measures of center do a good job as a typical measurement for the distribution.