Question

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

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

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

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 = \(\sqrt{\frac{0.2 \times (1-0.2)}{400}} = \sqrt{\frac{0.16}{400}} = 0.02\)

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.

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

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%.