1. Reading level count.
How many of these patients had a 6th grade reading level?
Solution.
3
Section 8.3 Investigation 2.3: Readability of Cancer Pamphlets
Exercises 8.3.1 The Study
Researchers in Philadelphia investigated whether pamphlets containing information for cancer patients are written at a level that the cancer patients can comprehend. They applied tests to measure the reading levels of 63 cancer patients and also the readability levels of 30 cancer pamphlets (based on such factors as sentence length and number of polysyllabic words). These numbers correspond to grade levels, but cancer patient reading levels below grade 3 and above grade 12 were not determined exactly.
The following tables indicate the number of patients at each reading level and the number of pamphlets at each readability level:
Patient’s Reading Level
| Patient’s reading level | < 3 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | > 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Count | 6 | 4 | 4 | 3 | 3 | 2 | 6 | 5 | 4 | 7 | 2 | 17 |
Pamphlet’s Readability Level
| Pamphlet’s readability level | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Count | 3 | 3 | 8 | 4 | 1 | 1 | 4 | 2 | 1 | 2 | 1 |
2. Limitation of calculating mean.
Explain why the form of the data does not allow you to calculate the mean reading level of these cancer patients.
Solution.
Because there are data groups such as < 3 and > 12 that prevent us from knowing the true distribution of the reading levels.
3. Median reading level for patients.
The median reading level is th grade.
Hint.
Consider the counts, and remember there are 63 patients. Find the middle value (32nd patient).
Solution.
The median reading level is 9th grade.
4. Median readability level for pamphlets.
The median readability level is th grade.
Hint.
With 30 pamphlets, find the average of the 15th and 16th values when arranged in order.
Solution.
The median readability level is 9th grade.
5. Comparing medians.
How do these medians compare?
-
The medians are the same (both 9th grade), so the values are very close.
-
Correct! The medians are identical.
-
The patient median is higher than the pamphlet median.
-
Incorrect. Check your calculations for both medians.
-
The pamphlet median is higher than the patient median.
-
Incorrect. Check your calculations for both medians.
-
The medians are several grades apart.
-
Incorrect. Calculate both medians carefully.
6. Interpreting closeness of medians.
Does the closeness of these medians indicate the pamphlets are well matched to the patients’ reading levels? Explain. How would you respond to these researchers?
Hint.
You may want to perform some additional explorations of these data.
Solution.
No, it does not; there could be differences in the shape of the data that make the two data groups not very well matched. In particular, we notice there are 17 patients who can’t read the simplest pamphlet.
Discussion.
Keep in mind when examining quantitative data, that you should start by constructing some simple graphs to explore the data. In fact, you should use technology to explore a couple of different graphs (e.g., change the bin width in a histogram) to help reveal hidden patterns and unusual observations.
Study Conclusions.
When these researchers came to the statistical consultant, they wanted a p-value for comparing the mean reading level of the patients to the mean reading level of the pamphlets. You will learn about such a test in Chapter 4. However, the statistical consultant replied that not only couldn’t means be calculated, but that a simple look at the data revealed a substantial percentage of the patients with a reading level below that of the simplest pamphlet. This "descriptive" analysis was sufficient, rather than looking for a more complicated "inferential" analysis involving p-values and confidence intervals.
Subsection 8.3.2 Practice Problem 2.3
Checkpoint 8.3.1. Estimating tax revenue.
If city officials wanted to estimate total tax revenue for a city (how much money people will pay in based on their incomes), would you rather know the mean income or the median income? Explain.
Checkpoint 8.3.2. Estimating housing cost.
If you want to estimate the typical housing cost for a new city, would it be more helpful to know the mean housing cost or the median housing cost? Explain.
Checkpoint 8.3.3. Research question focus.
In the Cancer Pamphlet study, explain why a comparison of the centers of the distributions does not match the research question of interest.
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