Question

A random sample will be selected from the population of all adult residents of a particular city. The sample proportion \(\hat{p}\) will be used to estimate \(p,\) the proportion of all adult residents who do not own a car. For which of the following situations will the estimate tend to be closest to the actual value of \(p ?\) i. \(\quad n=500, p=0.1\) $$ \text { ii. } \quad n=1,000, p=0.2 $$ iii. \(n=1,200, p=0.3\)

Short answer

Scenario ii. (n=1000, p=0.2) will tend to be the closest to the actual value of \( p \).

Step-by-step solution

Calculate Standard Error for Scenario 1

Initialize with \( n=500 \) and \( p=0.1 \). Insert these values into the formula for Standard Error thus: \( SE_{1} = \sqrt{ (0.1*(1-0.1))/500 } = 0.0134 \).

Calculate Standard Error for Scenario 2

Next calculate Standard Error for the second scenario with \( n=1000 \) and \( p=0.2 \). Insert these into the Standard Error formula: \( SE_{2} = \sqrt{ (0.2*(1-0.2))/1000 } = 0.01265 \).

Calculate Standard Error for Scenario 3

Finally, calculate the Standard Error for scenario 3 with \( n=1200 \) and \( p=0.3 \). Using the formula again we find: \( SE_{3} = \sqrt{ (0.3*(1-0.3))/1200 } = 0.0134 \).

Compare the Standard Errors

Now that we have the Standard Errors for each scenario, compare them to identify the lowest one. The scenario with the lowest Standard Error will produce the estimate that tends to be closest to the actual value of \( p \). From the calculations, \( SE_{2} \) which is 0.01265, is the lowest hence Scenario 2 ( ii. \( n=1,000, p=0.2 \)) will tend to give the most accurate estimate.

Key concepts

Sample Proportion

The sample proportion, often denoted as \(\hat{p}\), represents the fraction of individuals in a sample that exhibit a certain characteristic. For example, in a study to understand transportation habits, \(\hat{p}\) could indicate the proportion of people in a random sample of residents who do not own a car. It serves as the best estimate of the population proportion within the confines of the sample taken.

The accuracy of the sample proportion as an estimator for the actual population proportion relies heavily on the size of the sample and how representative it is of the population. If the sample is too small or not random, it might misrepresent the population, leading to errors in estimation. The exercise aims to identify which of several scenarios would more accurately estimate the true population proportion, given different sample sizes and assumed true proportions.

Population Proportion

The population proportion, denoted by \(p\), is the actual proportion of individuals in an entire population possessing a specific feature. Continuing with the transportation example, \(p\) would represent the exact fraction of all adult residents in a city that does not own a car. Unlike the sample proportion, the population proportion is a fixed value and does not change regardless of sample size or sampling methods used.

Estimating the population proportion is often the main goal of statistical studies, and it is why calculating sample proportions and understanding their variance are so significant. Estimators, like the sample proportion, are used to infer the population proportion, as it is usually impractical to measure the entire population.

Sample Size

Sample size, denoted as \(n\), is the number of individual observations included in a sample. It plays a critical role in statistical analyses because larger samples tend to yield more reliable estimates of the population characteristics. The principles of probability suggest that as the sample size increases, the sample proportion \(\hat{p}\) should get closer to the true population proportion \(p\).

In choosing the sample size, a balance must be struck between precision and resources. Larger sample sizes decrease the standard error, making it a key player in creating a smaller margin of error and tighter confidence intervals. This is exhibited in our exercise where different sample sizes impact the standard error of the estimate.

Confidence Interval

A confidence interval provides a range of values within which the true population parameter is expected to lie, considering a certain level of confidence (often 95%). It takes into account the sample proportion and the standard error to give an interval estimate rather than a single point estimate.

Calculating a confidence interval includes understanding the distribution of your estimator. A larger sample size and smaller standard error lead to a narrower confidence interval, implying a more precise estimate of the population proportion. In practice, while a single study might give a particular confidence interval, repeating the study numerous times under the same conditions will result in a confidence interval that contains the true population proportion in the specified percentage of studies (such as 95% of the time for a 95% confidence interval).