Question

In Exercises 3.51 to 3.56 , information about a sample is given. Assuming that the sampling distribution is symmetric and bell-shaped, use the information to give a \(95 \%\) confidence interval, and indicate the parameter being estimated. \(\hat{p}_{1}-\hat{p}_{2}=0.08\) and the margin of error for \(95 \%\) confidence is \(\pm 3 \%\).

Short answer

The 95% confidence interval for the difference between \(\hat{p}_{1}\) and \(\hat{p}_{2}\) is [0.05, 0.11]. The parameter estimated is the difference of the true proportions \(p_{1}\) and \(p_{2}\).

Step-by-step solution

Understanding the given information

Based on the information provided, we know that the difference between the sample proportions (\(\hat{p}_{1}\) and \(\hat{p}_{2}\)) is 0.08, and that the margin of error, E, at 95% confidence level is ±3% or 0.03.

Constructing the confidence interval

For a 95% confidence interval, we can construct the interval as follows: \((\hat{p}_{1}-\hat{p}_{2}) \pm E\), where E is the margin of error. In our case, we substitute the known values: \(0.08 \pm 0.03\)

Solving for the interval

To find the actual confidence interval, perform the addition and subtraction. This gives us two values, which will be the lower and upper bounds of the confidence interval, respectively.

Indicating the estimated parameter

The parameter we are estimating is the difference in true proportions \(p_{1} - p_{2}\). The estimated parameter (also known as a point estimate) was given as \(\hat{p}_{1}-\hat{p}_{2} = 0.08\). The 95% confidence interval gives us a range of values that this difference could take, given our data.

Key concepts

Sampling Distribution

Sampling distribution is a term used in statistics to describe the distribution of sample statistics, such as the mean or proportion, obtained from a large number of samples drawn from the same population. For example, imagine flipping a coin 50 times to find the proportion of heads — this is one sample proportion. If you repeat this process many times, each time with a new set of 50 flips, you'll get different sample proportions. The collection of these sample proportions is the sampling distribution of the sample proportion.

When the sample size is large, the central limit theorem states that the sampling distribution of the sample mean or proportion will be approximately normal or bell-shaped, regardless of the shape of the population distribution. The exercise provided involves the difference between two sample proportions, \(\hat{p}_1 - \hat{p}_2\), which will also have its own sampling distribution that we assume to be normal based on the exercise's context.

Margin of Error

The margin of error represents how much we expect our estimate to vary from the true population parameter. It is influenced by both the level of confidence we desire and the variability in the sample data.

In the context of a 95% confidence interval, the margin of error tells us that if we were to take many such samples and construct confidence intervals for each, about 95% of those intervals would contain the true parameter. With a margin of error of \(\text{\pm 3\%}\) or \(0.03\), there's an implied level of precision in our estimate of the difference between two population proportions. The calculation of the margin of error depends on the standard error of the statistic and the desired confidence level, which indicates how confident we are that the true parameter lies within the interval.

Sample Proportions

Sample proportions are a way of expressing the presence of a characteristic within a sample. For instance, if we surveyed 100 people on whether they like ice cream and 60 said yes, the sample proportion who like ice cream is \(\hat{p} = 0.60\). Sample proportions are used as estimates of the population proportions, from which the samples are drawn.

When conducting a study with two samples, we might be interested in the difference between their proportions, as seen in the textbook exercise. By comparing \(\hat{p}_1\) and \(\hat{p}_2\), we are looking at how much one sample's proportion differs from the other. This difference, along with an appropriate margin of error, helps form a confidence interval for the difference in population proportions.

Statistical Inference

Statistical inference involves making predictions or decisions about population parameters based on sample data. It encompasses various techniques, including hypothesis testing, estimating population parameters, and constructing confidence intervals, which is the focus of our given exercise.

With the provided information that the difference between the sample proportions is \(0.08\) and the margin of error is \(\text{\pm 3\%}\), we are inferring something about the true difference in proportions in the wider population from which the samples were taken. Statistical inference allows us to move from the sample, our known data, to the population, the broader context we're interested in, providing a range where we expect the true parameter lies with a certain degree of confidence.