Question

In response to budget cuts, county officials are interested in learning about the proportion of county residents who favor closure of a county park rather than closure of a county library. In a random sample of 500 county residents, 198 favored closure of a county park. For each of the three statements below, indicate if the statement is correct or incorrect. If the statement is incorrect, explain what makes it incorrect. Statement 1: It is unlikely that the estimate \(\hat{p}=0.396\) differs from the value of the actual population proportion by more than 0.0429 Statement 2: The estimate \(\hat{p}=0.396\) will never differ from the value of the actual population proportion by more than 0.0429 Statement 3: It is unlikely that the estimate \(\hat{p}=0.396\) differs from the value of the actual population proportion by more than 0.0219

Short answer

Fill the short answer after the calculations of standard error, margin of error, and evaluating each statement.

Step-by-step solution

Calculate standard error

First, we need to calculate the standard error of the proportion using the formula ~\( SE = \sqrt{ \hat{p} (1 - \hat{p}) / n} \). Substituting the given values, we get \( SE = \sqrt{ 0.396 * (1 - 0.396) / 500 } \).

Calculate margin of error

The margin of error (MOE) is usually approximated as 2 standard errors, so we should multiply our calculated standard error by 2 to obtain the margin of error. \( MOE = 2 * SE \).

Evaluate Statements 1 - 3

With the calculated margin of error, we can now evaluate each statement. If the specified value in each statement is within the margin of error from the sample proportion \( \hat{p} \), it is likely that the estimate \( \hat{p} \) could differ by that much from the actual population proportion. If not, the statement is incorrect. We evaluate each statement in turn.

Evaluate Statement 1

Statement 1 suggests that it is unlikely the estimate \( \hat{p} \) differs from the population proportion by more than 0.0429. If 0.0429 is within the margin of error calculated before, this statement is correct. Otherwise, it is incorrect.

Evaluate Statement 2

Statement 2 suggests that the estimate \( \hat{p} \) will never differ from the population proportion by more than 0.0429. The word 'never' makes this statement incorrect, because there is always a chance it could, even if it's within our calculated margin of error.

Evaluate Statement 3

Statement 3 suggests that it is unlikely the estimate \( \hat{p} \) differs from the population proportion by more than 0.0219. If 0.0219 is within the margin of error calculated before, this statement is correct. Otherwise, it is incorrect.

Key concepts

Standard Error

When estimating population proportions based on sample data, the standard error (SE) plays a crucial role. The standard error measures the variability or precision of the sample estimate, indicating how much the sample proportion can be expected to fluctuate from the true population proportion.

The formula to calculate the SE of a proportion is given by \( SE = \sqrt{ \hat{p} (1 - \hat{p}) / n } \) where \( \hat{p} \) is the sample proportion, and \( n \) is the sample size. A smaller SE indicates a more precise estimate, as the sample proportion is likely closer to the population proportion. Conversely, a larger SE implies more variability and a less reliable estimate.

Understanding SE helps in interpreting the reliability of the estimate: the larger the sample size, \( n \) , the smaller the SE, implying a more stable estimate of the population parameter. This is why increasing the sample size is a common method for reducing the uncertainty in our estimates.

Margin of Error

The margin of error (MOE) quantifies the range of values within which the true population parameter is expected to fall. It's crucial for understanding the level of uncertainty in a point estimate from a sample, allowing us to say how close the sample proportion is to the true population proportion.

The MOE is calculated by multiplying the standard error by a critical value that corresponds to a certain level of confidence (commonly the z-score for a desired confidence level). For simplicity, and assuming a 95% confidence interval, the critical value is often approximated as 2. Therefore, the MOE is approximated as \( MOE = 2 * SE \) .

A smaller MOE suggests that the sample proportion estimate is more precise, i.e., we can be more confident that the true population proportion lies within the calculated range around the sample proportion. A larger MOE implies a wider interval and hence less certainty about the proximity of the sample estimate to the population proportion.

The use of the term 'never' in relation to MOE, as seen in the exercise, is misleading. It suggests absolute certainty which is not possible in statistics; there's always a probability, however small, that the true parameter lies outside of the estimated range.

Sampling Distribution

The sampling distribution is a theoretical distribution of a statistic, like a mean or proportion, that we would obtain if we drew all possible samples of a given size from the population. It helps us to understand the behavior of the sample statistics and forms the basis for conducting statistical inference.

In the context of estimating a population proportion, the sampling distribution of the sample proportion can be approximated by a normal distribution, under the condition of a large enough sample size. This normal approximation is essential because it allows us to use the properties of the normal curve, such as specifying confidence intervals and testing hypotheses.

The Central Limit Theorem supports the use of the normal model for sampling distributions, stating that with a sufficiently large sample size, the sampling distribution of the sample mean will be approximately normal, regardless of the distribution in the population. A general rule of thumb is that this normal approximation is reasonable if both \( n\hat{p} \) and \( n(1-\hat{p}) \) are greater than 5.