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Chapter 20: Problem 30

The painful wrist condition called carpal tunnel syndrome can be treated with surgery or, less invasively, with wrist splints. Recently, Time magazine reported on a study of 176 patients. Among the half that had surgery, \(80 \%\) showed improvement after three months, but only \(48 \%\) of those who used the wrist splints improved. a. What's the standard error of the difference in the two proportions? b. Construct a \(95 \%\) confidence interval for this difference. c. State an appropriate conclusion.

Short Answer

Expert verified
First, calculate the standard error of the difference between the two proportions using the proportions and sizes for each group. Then calculate the difference in proportions. Use this information to construct the 95 percent confidence interval for the difference. Draw a final conclusion based on whether or not the confidence interval contains zero.

Step by step solution

01

Identify Proportions and Sizes

Identify the proportions and sizes for each group. If we call the surgery group 'group A' and the splints group 'group B', then:Proportion of improvement in group A, \(p_A = 0.80\).Number in group A, \(n_A = 0.50 \times 176 = 88\).Proportion of improvement in group B, \(p_B = 0.48\).Number in group B, \(n_B = 0.50 \times 176 = 88\).
02

Calculate Standard Error

Calculate the standard error of the difference in the two proportions. Using the formula:\[SE = \sqrt{\frac{{p_A\times (1-p_A)}}{{n_A}} + \frac{{p_B\times (1-p_B)}}{{n_B}}}\]Substitute the identified values:\[SE = \sqrt{\frac{{0.80\times (1-0.80)}}{{88}} + \frac{{0.48\times (1-0.48)}}{{88}}}\]
03

Calculate Difference in Proportions

Calculate the difference in proportions, which is \(p_A - p_B = 0.80 - 0.48 = 0.32\). This difference is needed for constructing the confidence interval in the next step.
04

Construct 95 percent Confidence Interval

Construct the 95 percent confidence interval for the difference using the formula \[(p_A - p_B) \pm 1.96 \times SE\]Substitute the calculated difference in proportions and standard error into the formula to calculate the confidence interval.
05

Conclusion Drawing

Draw a conclusion on whether or not there is a statistically significant difference in the proportions based on the calculated confidence interval.If the confidence interval does not contain zero, it suggests there is a significant difference between the two treatment methods.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Confidence Interval
A confidence interval gives us a range where we expect our true value to lie. It shows us how much uncertainty is around a measurement or estimate. For the case of carpal tunnel treatments, we're interested in the difference between the improvement rates of surgery and wrist splints.
The confidence level, often set at 95%, tells us how sure we are about the interval capturing the true difference. Here's why it's useful:
  • It tells us the precision of our estimate.
  • It helps us make decisions. If the interval doesn鈥檛 contain zero, we see a significant difference.
To find the confidence interval, we use the formula: \[(p_A - p_B) \pm 1.96 \times SE\] Where \(1.96\) is the z-value for 95% confidence. By computing this, we get a range for the difference in improvement rates.
Standard Error
The standard error measures the variability of a statistic. It's like a yardstick for how much variability there might be in our estimates from sample to sample. In our scenario of treating carpal tunnel syndrome, the standard error of the difference in proportions helps us understand how much the difference between the effectiveness of surgery and wrist splints could change by chance alone.
The formula for calculating the standard error (SE) of the difference between two proportions is: \[SE = \sqrt{\frac{{p_A \times (1 - p_A)}}{{n_A}} + \frac{{p_B \times (1 - p_B)}}{{n_B}}}\] This tells us how spread out the difference in proportions could be. A smaller SE indicates a more precise estimate, meaning less variation by chance.
Proportion Difference
The proportion difference is simply the difference between two proportions. In statistics, this measures how one group differs from another. For our carpal tunnel study, we are comparing two treatments: surgery and wrist splints.
This difference is calculated as:\[p_A - p_B = 0.80 - 0.48 = 0.32\] This result means that surgery leads to a higher improvement rate by 32% compared to splints. Understanding this helps in evaluating which treatment is more effective, guiding better medical decisions. Knowing the difference allows us to quantify the advantage of one treatment over another.
Statistical Significance
Statistical significance helps us decide whether an observed effect is real or could just be due to chance. In analyzing carpal tunnel treatments, determining whether the difference in improvement rates is statistically significant is crucial.
First, we look at the confidence interval. If the interval for the difference between surgery and splints does not include zero, the difference is considered statistically significant.
  • If zero is not in the interval, it implies that there is a real difference.
  • If zero is in the interval, we can't confidently say one treatment is better.
This conclusion informs doctors whether surgery or splints make a significant difference in treatment success rates. Statistical significance, therefore, boosts confidence in clinical decisions and can affect treatment recommendations.

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Most popular questions from this chapter

Researchers comparing the effectiveness of two pain medications randomly selected a group of patients who had been complaining of a certain kind of joint pain. They randomly divided these people into two groups, then administered the pain killers. Of the 112 people in the group who received medication A, 84 said this pain reliever was effective. Of the 108 people in the other group, 66 reported that pain reliever B was effective. a. Write a \(95 \%\) confidence interval for the percent of people who may get relief from this kind of joint pain by using medication A. Interpret your interval. b. Write a \(95 \%\) confidence interval for the percent of people who may get relief by using medication B. Interpret your interval. c. Do the intervals for \(A\) and B overlap? What do you think this means about the comparative effectiveness of these medications? d. Find a \(95 \%\) confidence interval for the difference in the proportions of people who may find these medications effective. Interpret your interval. e. Does this interval contain zero? What does that mean? f. Why do the results in parts \(c\) and e seem contradictory? If we want to compare the effectiveness of these two pain relievers, which is the correct approach? Why?

Egyptians Some archaeologists theorize that ancient Egyptians interbred with several different immigrant populations over thousands of years. To see if there is any indication of changes in body structure that might have resulted, they measured 30 skulls of male Egyptians dated from 4000 B.c.E. and 30 others dated from 200 B.C.E. (A. Thomson and R. Randall-Maciver, Ancient Races of the Thebaid, Oxford: Oxford University Press, 1905\()\) a) Are these data appropriate for inference? Explain. b) Create a \(95 \%\) confidence interval for the difference in mean skull breadth between these two eras. c) Do these data provide evidence that the mean breadth of males' skulls changed over this period? Explain. d) Perform Tukey's test for the difference. Do your conclusions of part c change? \({ }^{*}\) e) Perform a rank sum test for the difference. Do your conclusions of part c change?

Political pundits talk about the "bounce" that a presidential candidate gets after his party's convention. In the past 40 years, it has averaged about 6 percentage points. Just before the 2004 Democratic convention, Rasmussen Reports polled 1500 likely voters at random and found that \(47 \%\) favored John Kerry. Just afterward, they took another random sample of 1500 likely voters and found that \(49 \%\) favored Kerry. That's a two percentage point increase, but the pollsters claimed that there was no bounce. Explain.

A study published in the Archives of General Psychiatry examined the impact of depression on a patient's ability to survive cardiac disease. Researchers identified 450 people with cardiac disease, evaluated them for depression, and followed the group for 4 years. Of the 361 patients with no depression, 67 died. Of the 89 patients with minor or major depression, 26 died. Among people who suffer from cardiac disease, are depressed patients more likely to die than non-depressed ones? a. What kind of design was used to collect these data? b. Write appropriate hypotheses. c. Are the assumptions and conditions necessary for inference satisfied? d. Test the hypothesis and state your conclusion. e. Explain in this context what your \(\mathrm{P}\) -value means. \(f\). If your conclusion is actually incorrect, which type of error did you commit? g. Create a 95\% Cl. h. Interpret your interval. i. Carefully explain what "95\% confidence" means.

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