Investigation 4.3: Left-Handedness and Life Expectancy

Section 19.2 Investigation 4.3: Left-Handedness and Life Expectancy

In this investigation you will explore which factors impact the size of the p-values/our assessment of statistical significance when comparing two sample means.

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Exercises 19.2.1 The Study

Psychologist Stanley Coren has conducted several studies investigating the life expectancy of left-handers compared to right-handers, believing that the stress of being left-handed in a right-handed world leads to earlier deaths among the left-handers.

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Introduction to the left-handedness and life expectancy study

In one study Coren and Halpern (1991) sent surveys to thousands of next-of-kin of recently deceased southern Californians and asked whether the person had been right-handed or left-handed. They were very careful in how they collected their data. First, they consulted a bereavement counselor who suggested that they not contact anyone unless at least 9 months had passed since the death. The counselor also suggested that they make the contact as gentle as possible and not follow up or press people for responses. The researchers also decided that they would not contact next of kin if the death had been a result of murder or suicide or if the deceased was a child age 6 or younger. They received 987 replies and found that the average age of right-handed people at death was 75 years and for left-handed people it was 66 years.

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1. Study Design.

Is this an observational study or an experiment?

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Observational study
Experiment
Both
Neither

Solution.

Observational study because the researchers did not determine which people would be left-handed.

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2. Random Samples.

Did the researchers take a random sample of left-handers and an independent random sample of right-handers?

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Yes
No
Not really but close enough

Solution.

This is really a single random sample (among those willing to participate). Participants were then asked about their handedness (cross-classified) but because there is no connection between the left and right handers in this study, it is reasonable to consider them two independent random samples.

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3. Independence Assumption.

Is it reasonable to consider these samples independent?

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Solution.

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Even though the researchers did not take a random sample of left-handers and a separate random sample of right-handers, we are still willing to consider these samples as independent because the results for one group should have no effect on the results for the other group. In a situation with two independent random samples, we can apply the two-sample \(t\)-test to make inferences about the difference in the population means.

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4. Additional Information Needed.

The summary of the study reported the sample means; what additional information do we need to assess the statistical significance of their difference? Why is this information important?

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Solution.

Need to know the sample sizes of the two individual groups as well as how much variability there is in the ages at death (sample standard deviations).

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5. Hypotheses.

Let \(\mu_L\) represent the mean lifetime for the population of left-handers and let \(\mu_R\) represent the mean lifetime for the population of right-handers. State the null and alternative hypotheses for Coren and Halpern’s study (based on the research conjecture).

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Null hypothesis: \(H_0\text{:}\)

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Alternative hypothesis: \(H_a\text{:}\)

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Solution.

\(H_0: \mu_L = \mu_R\) (left-handers and right-handers have the same mean lifetime)

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\(H_a: \mu_L < \mu_R\) (left-handers have a shorter mean lifetime than right-handers)

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The table below lists some guesses for the sample size breakdown (based on estimates of the proportion of the population that are left-handed) and the sample standard deviations.

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Scenario	Group	Sample size	Sample mean	Sample SD
1	left	99 (10% of 987)	66	15
	right	888	75	15
2	left	50 (5% of 987)	66	15
	right	937	75	15
3	left	50 (5% of 987)	66	25
	right	937	75	25
4	left	10 (1% of 987)	66	25
	right	977	75	25
5	left	99 (10% of 987)	66	50
	right	888	75	50

6. Comparing Scenarios: Effect of Standard Deviation.

Scenarios 2 and 3 have the same sample means and sample sizes but different standard deviations (SD = 15 vs. SD = 25). Which scenario will produce stronger evidence against the null hypothesis (lower p-value)? Match each scenario to the strength of evidence.

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Scenario 2 (SD = 15)
Stronger evidence (lower p-value)
Scenario 3 (SD = 25)
Weaker evidence (higher p-value)

Solution.

Scenario 2 (SD = 15) provides stronger evidence because less variability within groups makes it easier to detect a difference between groups.

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7. Comparing Scenarios: Effect of Sample Size.

Scenarios 3 and 4 have the same sample means and standard deviations but different sample sizes for left-handers (\(n_L\) = 50 vs. \(n_L\) = 10). Which scenario will produce stronger evidence against the null hypothesis (lower p-value)? Match each scenario to the strength of evidence.

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Scenario 3 (\(n_L\) = 50)
Stronger evidence (lower p-value)
Scenario 4 (\(n_L\) = 10)
Weaker evidence (higher p-value)

Solution.

Scenario 3 (\(n_L\) = 50) provides stronger evidence because a larger sample size in the smaller group reduces the standard error, making it easier to detect a difference.

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8. Hypothesis Tests for Multiple Scenarios.

See technology instructions in Investigation 4.2 to carry out the two-sample \(t\)-tests for these hypotheses, using the sample sizes, sample means, and sample standard deviations given in the table above. For each of these five scenarios, report the resulting standardized statistic, p-value, and whether or not the difference is statistically significant with a significance level of \(\alpha = 0.01\) in the table below.

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Scenario	\(t\)-statistic	p-value	Significant at 1% level?
1
2
3
4
5

Solution.

Scenario	\(t\)-statistic	p-value	Significant at 1% level?
1	-5.66	0.000	Yes
2	-4.13	0.000	Yes
3	-2.48	0.00813	Yes, barely
4	-1.13	0.144	No
5	-1.70	0.0449	No

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9. Effects of Sample Size and Standard Deviation.

Summarize what your analysis reveals about the effects of the sample size breakdown and the sample standard deviations on the values of the \(t\)-statistic and p-value.

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Solution.

When the sample size for the left handers is larger, we have more evidence against the null hypothesis (larger |t-statistics|, smaller p-values). When the sample standard deviations are larger, we have less evidence against the null hypothesis.

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10. Most Realistic Scenario.

Considering the five scenarios of sample sizes and standard deviations used in the table, which scenario do you think is most realistic for this study of left- and right-handers’ lifetimes? Explain, based on the context for these data.

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Solution.

Probably scenario 1 or 2 as they have more a more realistic percentage of left-handers and the sample standard deviations (15) are more reasonable (the others are too large if we are expecting about 32% of data values to fall more than one standard deviation above or below the mean. We probably aren’t expecting a normal distribution, but these standard deviations still feel too large).

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11. Causation vs. Association.

Based on your analyses and how these data were collected, are you convinced that being left-handed causes individuals to die sooner on average than being right-handed? How should you phrase your conclusion?

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Solution.

For even the remotely realistic scenarios, the p-values were quite small indicating statistical significance. However, we can’t draw any "cause-and-effect" conclusions as this was an observational study.

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12. Alternative Explanation.

Part of the motivation for this research was an earlier study by Porac and Coren (1981) that surveyed 5147 men and women of all ages in North America. They found that 15% of ten-year-olds were left-handed, compared to only 5% of fifty-year-olds and less than 1% of eighty-year-olds. At the age of 85, right-handers outnumbered left-handers by a margin of 200 to 1. Suggest another explanation for these puzzling findings that actually provides a counter-argument to the conclusion from the 1991 Coren and Halpern study that left-handers tend to live shorter lives than right-handers.

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Solution.

For those who would be in their eighties in 1981, many of them would have been encouraged to not be left-handed when they were younger. This would explain why there were fewer left-handers in the older age groups.

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Study Conclusions.

The difference in the sample mean lifetimes does appear to be statistically significant for all reasonable choices of the sample sizes and the sample standard deviations, so we have strong evidence that left-handers do tend to have shorter life spans than right-handers. However, there are numerous cautions to heed when drawing conclusions from such a study. This was a retrospective study with voluntary response, and in fact the researchers reported that they tended to hear from the left-handed relatives more often than right-handers, and they only heard from fewer than half of the families contacted. There is also no information given about the proportion of left-handers in the two southern California counties studied or the average ages of their residents now. In particular, one explanation for the lower percentage of left-handers among the elderly is not that they have died younger but that it used to be quite common practice to strongly encourage left-handed children to switch to being right-handed. Nowadays, that is less common (in fact many athletes love having this advantage!), and so now there is a higher percentage of left-handers among younger age groups. This helps to explain why the left-handers who had died would tend to be younger. In other words, maybe the average age difference between living left-handers and right-handers is also nine years.

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These studies have actually become hot topics for debate as some other studies have not been able to replicate Coren and Halpren’s results. Other prospective longitudinal studies in the United States (Marks and Williams, 1991; Wolf, D’Agostino, and Cobb, 1991) have not found a significant difference in age at death. Still others have found connections between handedness and accident rates, lower birth rates, cancer, alcohol misuse, smoking, and schizophrenia. Alas, we don’t see any randomized, comparative experiments being conducted to answer these questions in the near future!

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Discussion.

You should have found that larger variability in each sample (so a larger SE) produces a larger p-value and therefore less convincing evidence that the population means differ. Or to put this in a more positive light: Reducing variability within groups makes it easier to distinguish between the groups. You should also have found that a bigger discrepancy in sample sizes between the two groups produces a larger p-value and therefore less convincing evidence that the population means differ.

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Subsection 19.2.2 Practice Problem 4.3A

Checkpoint 19.2.1. Effect of Larger Mean Difference.

If all else stays the same but the difference in the means between the groups is larger, will the p-value be larger or smaller?

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Solution.

The greater the difference between the population means, the smaller the p-value.

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Checkpoint 19.2.2. Effect of Larger Sample Sizes.

If all else stays the same but the sample sizes within the groups are larger, will the p-value be larger or smaller?

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Solution.

The larger and more similar the sample sizes, the smaller the p-value.

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Subsection 19.2.3 Practice Problem 4.3B

Recall the study on children’s television viewing habits for two schools in San Jose, CA from Practice Problem 2.6B (Robinson, 1999). The researcher wanted to study whether a new classroom curriculum could reduce children’s television viewing habits, which might in turn help to prevent obesity. One of the schools, chosen at random, incorporated an 18-lesson, 6-month classroom curriculum designed to reduce watching television and playing video games, whereas the other school made no changes to its curriculum. Both before the curriculum intervention, all children were asked to report how many hours per week they spent on these activities.

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The following summary statistics pertain to the reports of television watching, in hours per week, prior to the intervention:

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Baseline	Sample size	Sample mean	Sample SD
Control group	103	15.46	15.02
Intervention group	95	15.35	13.17

Approximate two-sided p-value: 0.956

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Checkpoint 19.2.3. Baseline p-value Interpretation.

Do you think the researchers are happy or unhappy about the size of this p-value? Explain.

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Solution.

The researchers are happy about the large p-value at baseline because it indicates there is no significant difference between the control and intervention groups at the start of the study, which is what they want.

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Checkpoint 19.2.4. Alternative Baseline Scenario 1.

Suppose that the summary statistics at baseline had instead been the following:

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Baseline	Sample size	Sample mean	Sample SD
Control group	103	14.46	15.02
Intervention group	95	8.80	13.17

Without performing any calculations, how should the p-value for these data compare (larger or smaller) to the p-value from the actual baseline results? Explain.

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Solution.

The p-value would be smaller because the difference in means is larger (14.46 - 8.80 = 5.66 vs. 15.46 - 15.35 = 0.11).

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Checkpoint 19.2.5. Alternative Baseline Scenario 2.

Suppose that the summary statistics at the beginning of the study had instead been the following:

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Baseline	Sample size	Sample mean	Sample SD
Control group	103	15.46	13.82
Intervention group	95	15.35	10.41

Without performing any calculations, how should the p-value for these data compare (larger or smaller) to the actual baseline results? Explain.

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Solution.

The p-value would be smaller because the smaller the value of σ (sample standard deviations), the smaller the p-value.

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Checkpoint 19.2.6. Subsetting by Socio-Economic Class.

Suppose the researchers decide to look at a subset of children in this study that belong to the same social-economic class (with the expectation that their television watching habits will be more similar to each other). Discuss one advantage and one disadvantage to this approach in terms of detecting a difference between the control group and the intervention group at the conclusion of the study. [Hints: Relate to your answer in the previous question?]

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Solution.

Advantage: By reducing the variability within groups (smaller SDs), it would be easier to detect a difference between groups at the conclusion. Disadvantage: By subsetting, the sample sizes would be smaller, making it harder to detect a difference.

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