Question

In Exercises 6.1 to \(6.6,\) if random samples of the given size are drawn from a population with the given proportion, find the standard error of the distribution of sample proportions. Samples of size 60 from a population with proportion 0.90

Short answer

The standard error of the distribution of sample proportions for samples of size 60 from a population with a proportion of 0.90 is approximately 0.038.

Step-by-step solution

Identify the given values

From the problem, we're given that the sample size (n) is 60 and the population proportion (p) is 0.90.

Substitute the values into the standard error formula

Plugging these values into the standard error formula, we have \(SE = \sqrt{\frac{0.90(1-0.90)}{60}}\).

Perform the calculations

First, calculate the expression in the numerator. Since \(1 – 0.90 = 0.10\), the expression within the square root becomes \(0.90 * 0.10 / 60 = 0.0015\). The square root of 0.0015 is approximately 0.03782.

Key concepts

Sample Proportions

Understanding sample proportions is crucial when delving into statistical studies. The sample proportion, commonly represented as \( \hat{p} \), is an estimate of the population proportion \( p \) derived from a sample taken from the population. Consider it as a snapshot of what you're trying to measure across the entire population. For instance, if you want to know the percentage of left-handed students in a school, you could take a sample and find the proportion of left-handed students in that sample. This proportion is your sample proportion.

It's important to note that the sample proportion can vary from sample to sample. This variability is due to the random nature of sample selection. Ideally, the sample should be representative of the population to ensure that the sample proportion is a good estimate of the population proportion. Ensure this representation by using random sampling methods and a sufficient sample size to mitigate the risk of bias and decrease the margin of error.

Population Proportion

The population proportion represents the proportion of individuals in the entire population that have a particular attribute. It is denoted by \( p \) and is a fixed value, although in practice it's almost always unknown and needs to be estimated from the sample data.

In the context of the given exercise, the population proportion of 0.90 indicates that if we were to look at the entire population, we would expect 90% of the observations to have the characteristic of interest. This is a key parameter in many statistical procedures and is fundamental to hypothesis testing and confidence interval estimation. For accuracy in these procedures, it's critical to have a good estimate of the population proportion, which brings us back to the importance of sample proportions as an estimate of \( p \).

Standard Error Formula

The standard error (SE) is a statistical term that measures the accuracy with which a sample distribution represents a population using standard deviation. In the context of sample proportions, the standard error formula is expressed as \( SE = \sqrt{\frac{p(1 - p)}{n}} \), where \( p \) is the population proportion and \( n \) is the sample size.

Using the Standard Error Formula

In our exercise, you've been given a sample size of 60 and a population proportion of 0.90. By substituting these values into the formula, you calculate the SE to assess how much the sample proportion is expected to vary from the true population proportion. The resulting standard error lets us gauge the potential error or uncertainty in our sample estimate, enabling us to understand the reliability of our sample proportion and to construct confidence intervals around it.

A key factor to remember is that the standard error decreases as the sample size increases, meaning that larger samples tend to yield more precise estimates of the population proportion. Conversely, a smaller sample size would increase the standard error, suggesting a less reliable estimate. This inverse relationship is fundamental in designing studies and interpreting statistical data.