Question

A USA Today snapshot reported the results of a random sample of 500 women who were asked what reasons they might have to consider dieting, \({ }^{5}\) with the following results. a. Find a \(95 \%\) confidence interval for the proportion of all women who would consider dieting to improve their health. b. Find a \(90 \%\) confidence interval for the proportion of all women who would consider dieting in order to have more energy.

Short answer

Based on the given information, it is not possible to calculate the confidence intervals for the proportion of women who consider dieting for the given reasons. We need the number of women who consider dieting to improve their health and to have more energy in the sample. Please provide these numbers to proceed with the calculation.

Step-by-step solution

Calculate sample proportions

In a random sample of 500 women, we need to find the number of women who would consider dieting to improve their health and the number of women who would consider dieting to have more energy. These numbers are not provided in the exercise, so we cannot proceed to further steps without that information. Please provide the number of women who consider dieting for each reason.

Key concepts

Understanding Statistics

Statistics is a field of mathematics concerned with collecting, analyzing, interpreting, presenting, and organizing data. In the context of the exercise, statistics helps to estimate the characteristics of a broader population (all women considering dieting) using a sample (500 women surveyed). A key component of this process is the concept of a confidence interval, which provides a range of values that likely includes the true population parameter (in this case, the true proportion of all women who would consider dieting for health improvements or more energy).

Creating a 95% or 90% confidence interval implies that if we were to take many random samples and calculate a confidence interval for each, about 95% or 90%, respectively, would contain the true population proportion. This does not mean there's a 95% or 90% chance that any given interval contains the true proportion; it's about the long-term frequency of these intervals capturing the true value if the sampling were repeated under the same conditions.

Grasping Probability Basics

Probability is the measure of how likely an event is to occur out of the number of possible outcomes. It plays a crucial role in calculating confidence intervals; statisticians use probability distributions to determine the likely position of the population measure based on sample data. In our exercise, the use of probability helps to determine how confident we can be that the interval would include the true proportion of the total population.

Moreover, the level of confidence chosen for an interval, like 95% or 90%, is directly related to probability. These percentages are derived from the probability distributions that underlie the data—commonly the standard normal (Z) distribution or the Student's t-distribution, depending on the sample size and known or unknown population standard deviation.

Sample Proportions Explained

The sample proportion is a fraction that indicates the number of instances of a particular outcome in a sample relative to the total number of observations. In the exercise, it would be the number of women who answered they would consider dieting for a particular reason, divided by the total number of women in the sample (500 in this case).

For instance, if 300 out of 500 women said they would consider dieting to improve health, the sample proportion \( p \) of women considering dieting for health is \(\frac{300}{500} = 0.6\). This sample proportion is an estimate of the true population proportion. When we calculate confidence intervals, we're essentially saying 'We are X% confident that the true population proportion of women who would consider dieting for this reason falls within this interval', based on our sample proportion and the associated probability distributions.