1. Simulation Plan.
(a) Outline a simulation appropriate for this study design.
Section 24.4 Investigation 5.3: Near-Sightedness and Night Lights (cont.)
Exercises The Study
Recall
[cross-reference to target(s) "investigation-3-2" missing or not unique], where we examined a simplified version of the Quinn, Shin, Maguire, and Stone (1999) study of childhood lighting exposure and eye refraction. Here we use three categories for each variable.
| Dark | Night light | Room light | Total | |
|---|---|---|---|---|
| Far-sighted | 40 | 39 | 12 | 91 |
| Normal | 114 | 115 | 22 | 251 |
| Near-sighted | 18 | 78 | 41 | 137 |
| Total | 172 | 232 | 75 | 479 |
Here we can view the data as one random sample cross-classified by lighting type and eye condition.
\(H_0\text{:}\) no association between lighting type and eye condition in the population.
\(H_a\text{:}\) there is an association.
This is a chi-squared test of association.
When technical conditions are met, we generally use the chi-squared theoretical null model.
2. Expected Count Calculation.
(b) Use the general formula to calculate the expected count for the Room light and Far-sighted cell.
3. Check Conditions.
(c) Is the chi-squared distribution valid for this table? Explain.
4. Technology Output and Contributions.
(d) Use technology to calculate the chi-squared statistic, verify degrees of freedom, and find the p-value. Also examine cell contributions and identify where the largest discrepancies occur.
5. Conclusions Paragraph.
(e) Write a paragraph summarizing conclusions, including statistical significance, scope of generalization, and causation.
Study Conclusions.
The segmented bar graph suggests near-sightedness increases with lighting level in this sample. A chi-squared test of association is appropriate because expected counts are sufficiently large. The p-value is extremely small, indicating strong evidence of association between lighting and eye condition in the population represented by this sample. The largest cell discrepancies are fewer near-sighted children than expected in the dark group and more than expected in the room-light group.
Because this is an observational study, a cause-and-effect conclusion is not warranted. Confounding variables (for example, parental vision and related household lighting choices) may explain part of the observed association. Generalization should also be cautious because children were not sampled as a simple random sample from all children.
6. Practice Problem 5.3.
The following table classifies births in 2002 by mother’s race and gestation length:
| White (non-Hispanic) | Black (non-Hispanic) | Hispanic | |
|---|---|---|---|
| Pre-term (under 37 weeks) | 251132 | 101423 | 99510 |
| Full term (37-42 weeks) | 1885189 | 435923 | 692314 |
| Post-term (over 42 weeks) | 149898 | 36896 | 64997 |
(a) Conduct a chi-squared test of association and report hypotheses, conditions, statistic, p-value, and conclusion.
(b) Which two or three cells contribute most to \(\chi^2\text{?}\) Are observed counts above or below expected in those cells, and what does that reveal?
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