Math 565 Homework #5
- Averages. Here are three commonly used measures for the central tendency of a set of numerical data items:
• Mean: Add the items and divide by the number of items.
• Median: Arrange the items in numerical order and locate the item “in the middle” (or if there are two “middle” items, then take their average).
• Mode: The item that occurs most often.
Suppose a student got these grades on different tests. (Each score is out of a maximum of 100 points.)
80, 90, 85, 90, 50, 78, 84
a. Find the value of each of the three measures of central tendency for this set of numbers.
b. Suppose you needed to assign a single number as a grade for this student. Discuss the strengths and weaknesses of each of the three measures of central tendency as a method of choosing this number. Also state what number you would actually choose. (You can choose one of the answers to part (a) or something different from any of them.) Justify your choice
- Bias. In everyday language, the word “bias” is often used to mean an unfair judgment, especially against a particular racial or ethnic group. In polling, it refers to
a built-‐‑in imbalance in the sampling process, which may occur without any malicious intent. Thus, a poll with a biased sample might not give correct information about the larger population, because it may slant the results in a certain direction, even if the pollster doesn’t have that intention. In this assignment, you will look at ways in which bias might enter into the polling process for a particular situation.
Suppose a student government committee wants to know whether $45 per ticket is too much to charge for a homeless shelter benefit concert at college. The committee decides to poll some students and use the results to help them determine if that price is too high.
Explain what might be wrong with each of these following methods of choosing a sample. How might it bias the results? What incorrect impression might we get?
a. Picking every tenth student who drives into the parking garage.
b. Stopping several groups of students coming out of the student union together and asking everyone in each group.
c. Picking one mathematics class at random and polling all the students in that class.
- Representative Samples. When you want to test hypotheses about some population, we can’t usually test the entire population. You usually need to pick a sample that is likely to represent the population accurately. Consider each situation below and answer the questions.
a. A music producer wants to find out what college students think about various types of popular music. The producer conducts a survey around the video games area of the student union. What is an example of a conclusion the producer might reach based on this survey that might not be true about college students in general?
b. An auto manufacturer wants to conduct on-the-street interviews to find out what adults in the United States think of the company’s latest TV advertising campaign. The interviewer decides to use the people standing at the bus stop near her home throughout the day as the sample population.
What is an example of an erroneous conclusion one might draw from this population?
What might be a more representative sample of the audience targeted by the advertisement?
- Your own survey. When we talk about a population in common English, it usually refers to human beings. In statistics, it can refer to any set of objects. For instance, we might wonder about a feature of shirts or plants.
We’ll use our own environment such as our classes, work or home to gather numerical information from objects (not necessarily people). This should result in twenty numbers, each the represents some feature of the data. For example, I could record the number of buttons on 20 shirts at my house as my data.
a. Decide what numerical data to collect, present the data you gathered, and describe your process for gathering it.
b. What are the mean, median, mode of your data? Do any of the averages express a “typical” element of your data? Explain why or why not.
c. To draw a conclusion from my data that is a statement about some population I have to be careful. If I collected the numbers of buttons on 20 shirts around my house, those numbers may not be typical of all shirts in the universe. First of all, Other countries might have shirts with different numbers of buttons. I might prefer one type of shirt or shirts for women (like me) might have a different number of buttons than shirts for men. So I can only draw a conclusion about a population for which my sample is likely to be representative.
State a conclusion you could make about your population based on the sample data you collected. For example, I might say, “Shirts for American women have a mean number of buttons equal to 6.5.” Explain why you think the sample is representative of your population.
- Representations of Data. Here is a list of midterm grades (out of 100) for a professor.
77, 96, 58, 100, 66, 76, 88, 73, 94, 75, 76, 84, 91, 74, 87, 92, 67
Here are a few different representations of the data along with the grades she assigned to
a. Explain how she decided which numerical score received which letter grade.
b. If another student were to receive an 81 on the midterm, what grade do you think she would receive and why?
c. Explain, so that a middle school student could understand, how to interpret each of the four representations and exactly how you would construct each of the charts from the data.
Note: To figure out the last one you need to know it is based on dividing the data into four groups with equal numbers of data points.
d. Explain one advantage each representation has over the other three.