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Chapter 8: Problem 26

A researcher classified his subjects as right-handed or left-handed 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 lefthanded using a \(95 \%\) confidence interval.

Short Answer

Expert verified
Answer: The 95% confidence interval for the proportion of left handed males in the entire population is (0.1608, 0.2392), indicating an estimated range of 16.08% to 23.92%.

Step by step solution

01

Calculate sample proportion and z-score

We are given that there are 80 left handed males in a sample of 400. So we can calculate the sample proportion (\(\hat{p}\)) as: \(\hat{p} = \frac{80}{400} = 0.2\) For a 95% confidence interval, the z-score (\(Z\)) from the standard normal distribution is approximately 1.96.
02

Calculate the confidence interval for the population proportion

Now, we will use the formula for the confidence interval of population proportion: \(CI = \hat{p} \pm Z \cdot\sqrt{ \frac{\hat{p}(1-\hat{p})}{n} }\) Plugging in the values we have: \(CI = 0.2 \pm 1.96 \cdot\sqrt{ \frac{0.2(1-0.2)}{400} }\) \(CI = 0.2 \pm 1.96 \cdot\sqrt{ \frac{0.16}{400} }\) \(CI = 0.2 \pm 1.96 \cdot 0.02\) \(CI = 0.2 \pm 0.0392\) Thus, the confidence interval for the population proportion is \((0.1608, 0.2392)\). This indicates that with 95% confidence, we estimate that between 16.08% and 23.92% of all males in the population would be classified as left handed according to the researcher's criterion.

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Key Concepts

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

Sample Proportion
Understanding the concept of the sample proportion is essential when dealing with statistics, especially when trying to estimate features of an entire population. In the context of our exercise, the sample proportion (\( \text{represented as} \) \( \(\hat{p}\) \) ) refers to the percentage of individuals within a sample meeting a specified criterion. In this case, it's the fraction of left-handed men among the 400 observed.

To calculate the sample proportion, we simply divide the number of 'successes' (left-handed men) by the total number of observations (400 men): \[ \hat{p} = \frac{\text{Number of left-handed men}}{\text{Total number of men}} \]. The result, 0.2 or 20%, represents the estimated proportion of left-handed males in the sample, which serves as a stepping stone to infer about the population proportion.
Z-Score
A z-score is a statistical metric that describes a data point's relationship to the mean of a group of values, measured in terms of standard deviations. When constructing confidence intervals for population proportions, the z-score helps determine how 'confident' we wish to be about our estimate.

In our example, a 95% confidence interval corresponds to a z-score of approximately 1.96. This value tells us how many standard deviations away from the mean we extend on both sides of the sample proportion to capture the true population proportion in 95% of samples. This z-score is derived from the standard normal distribution, which is a theoretical distribution that allows statisticians to understand probabilities and critical values associated with normally distributed data.
Population Proportion
The population proportion represents the true percentage of individuals in the entire population who meet a certain criterion. Unlike the sample proportion, which is a point estimate taken from a subset, the population proportion is often not known and must be estimated.

To infer about the population proportion based on a sample, we use the sample proportion and compute a confidence interval. This interval gives us a range of values that, with a certain level of confidence (in this case, 95%), contains the true population proportion. The width of this interval is influenced by the size of the sample and the variance within the sample.
Standard Normal Distribution
The standard normal distribution, also known as the z-distribution or Gaussian distribution, plays a pivotal role in statistics, especially when modeling random variables and conducting hypothesis testing. It's a special case of the normal distribution with a mean of zero and a standard deviation of one.

What makes it so powerful is that it allows statisticians to make probabilistic statements about data points by converting any normal distribution to a standard normal distribution using z-scores. For instance, when we use a z-score of 1.96 in our exercise, we are utilizing the properties of the standard normal distribution to state that the true population proportion lies within a certain interval around the sample proportion, with a 95% level of certainty.

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Most popular questions from this chapter

Acid rain-rain with a pH value less than \(5.7,\) caused by the reaction of certain air pollutants with rainwater-is a growing problem in the United States. Suppose water samples from 40 rainfalls are analyzed for \(\mathrm{pH}\), and \(\bar{x}\) and \(s\) are equal to 3.7 and .5 , respectively. Find a \(99 \%\) confidence interval for the mean \(\mathrm{pH}\) in rainfall and interpret the interval. What assumption must be made for the confidence interval to be valid?

Calculate the margin of error in estimating a binomial proportion \(p\) for the sample sizes given in Exercises \(11-14\). Use \(p=.5\) to calculate the standard error of the estimator, and comment on how an increased sample size affects the margin of error. \(n=400\)

Explain what is meant by "margin of error" in point estimation.

Independent random samples are drawn from two quantitative populations, producing the sample information shown in the table. Find a \(95 \%\) upper confidence bound for the difference in the two population means. $$\begin{array}{lcc}\hline & \multicolumn{2}{c} {\text { Population }} \\\\\cline { 2 - 3 } & 1 & 2 \\\\\hline \text { Sample Size } & 50 & 50 \\\\\text { Sample Mean } & 12 & 10 \\\\\text { Sample Standard Deviation } & 5 & 7 \\\\\hline\end{array}$$

Suppose that you were designing a research poll that included questions about illegal immigration into the United States, and the federal and state responses to the problem. a. Explain how you would select your sample. What problems might you encounter in this process? b. If you wanted to estimate the percentage of the population who agree with a particular statement in your survey questionnaire correct to within \(1 \%\) with probability \(.95,\) approximately how many people would have to be polled?
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