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Chapter 8: Problem 113

A researcher classified his subjects as innately right-handed or lefthanded by comparing thumbnail widths. He took a sample of 400 men and found that 80 men could be classified as left-handed according to his criterion. Estimate the proportion of all males in the population who would test to be left-handed using a \(95 \%\) confidence interval.

Short Answer

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Question: Estimate the proportion of all males in the population who would test to be left-handed using a 95% confidence interval based on the given sample data: 80 out of 400 males are left-handed. Answer: Based on the researcher's criterion for classifying left-handed males, we can estimate with 95% confidence that the proportion of left-handed males in the population lies between 16.08% and 23.92%.

Step by step solution

01

Calculate the sample proportion of left-handed males

To calculate the sample proportion, we need to divide the number of left-handed males by the total number of males in the sample: Sample proportion, p = (Number of left-handed males) / (Total number of males) p = 80 / 400 = 0.2
02

Calculate the standard error of the proportion

To calculate the standard error of the proportion, we will use the following formula: Standard error(SE) = \(\sqrt{\frac{p \times (1-p)}{n}}\) Where, \(p\) = Sample proportion \(n\) = Total number of males in the sample SE = \(\sqrt{\frac{0.2 \times (1-0.2)}{400}} = \sqrt{\frac{0.16}{400}} = 0.02\)
03

Determine the z-score for a 95% confidence interval

A 95% confidence interval corresponds to z-scores of \(\pm 1.96\). This is because about 95% of the area under the standard normal distribution lies within \(\pm 1.96\) standard deviations of the mean.
04

Calculate the confidence interval

To calculate the 95% confidence interval, we will use the following formula: Confidence interval = Sample proportion \(\pm\) (z-score x Standard error) 95% Confidence interval = 0.2 \(\pm\) (1.96 x 0.02) Lower bound = 0.2 - (1.96 x 0.02) = 0.1608 Upper bound = 0.2 + (1.96 x 0.02) = 0.2392
05

Interpret the results

Based on the researcher's criterion for classifying left-handed males, we can estimate with 95% confidence that the proportion of left-handed males in the population lies between 16.08% and 23.92%.

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Most popular questions from this chapter

Exercise 8.106 presented statistics from a study of fast starts by ice hockey skaters. The mean and standard deviation of the 69 individual average acceleration measurements over the 6 -meter distance were 2.962 and .529 meters per second, respectively. a. Find a \(95 \%\) confidence interval for this population mean. Interpret the interval. b. Suppose you were dissatisfied with the width of this confidence interval and wanted to cut the interval in half by increasing the sample size. How many skaters (total) would have to be included in the study?

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