Problem 25 - Basic Statistics for Business & Economics

Question

Suppose the U.S. president wants to estimate the proportion of the population that supports his current policy toward revisions in the health care system. The president wants the estimate to be within . 04 of the true proportion. Assume a \(95 \%\) level of confidence. The president's political advisors found a similar survey from two years ago that reported that \(60 \%\) of people supported health care revisions. a. How large of a sample is required? b. How large of a sample would be necessary if no estimate were available for the proportion supporting current policy?

Step-by-Step Solution

Answer

a. 577; b. 601.

Step 1: Understanding the Problem

We want to calculate the sample size required to estimate the proportion of the population that supports a policy, within a margin of error (E) of 0.04, at a confidence level of 95%. We have two scenarios: one with a previous estimate of the population proportion, and another without.

Step 2: Determine the Critical Value

For a 95% confidence level, we use the standard normal distribution. The critical value (Z) is 1.96. This can be found from Z-tables or standard normal distribution calculators.

Step 3: Scenario a: Using the Previous Estimate

The formula for sample size when an estimate of the population proportion (\( p \)) is available is: \[ n = \left( \frac{Z^2 \cdot p \cdot (1-p)}{E^2} \right) \]Substitute \( p = 0.60 \), \( Z = 1.96 \), and \( E = 0.04 \):\[ n = \left( \frac{1.96^2 \cdot 0.60 \cdot 0.40}{0.04^2} \right) \]

Step 4: Calculate Sample Size with Estimate

Calculating:\[ n = \left( \frac{3.8416 \cdot 0.24}{0.0016} \right) = \left( \frac{0.921984}{0.0016} \right) = 576.24 \]Rounding up since sample size must be a whole number, we get \( n = 577 \).

Step 5: Scenario b: Without the Previous Estimate

The formula for sample size without an estimate is:\[ n = \left( \frac{Z^2 \cdot 0.25}{E^2} \right) \]This uses \( p = 0.50 \) for maximum variability. Substitute \( Z = 1.96 \) and \( E = 0.04 \): \[ n = \left( \frac{1.96^2 \cdot 0.25}{0.04^2} \right) \]

Step 6: Calculate Sample Size without Estimate

Calculating:\[ n = \left( \frac{3.8416 \cdot 0.25}{0.0016} \right) = \left( \frac{0.9604}{0.0016} \right) = 600.25 \]Rounding up, we get \( n = 601 \).

Key Concepts

Margin of Error

The margin of error is a crucial element when it comes to estimating a population parameter, like a proportion. In simple terms, it reflects how close we think an estimate will be to the actual population value. Margins of error are often expressed as a percentage. They help us understand the potential difference between our estimate and the true value.

When you see a margin of error stated with a plus or minus symbol, like ±4%, it means that the true value could be 4% more or less than what your estimate is. For example, if 60% of a sample supports a policy, with a 4% margin of error, the true support is likely between 56% and 64%.

The margin of error allows for a buffer. It accounts for sampling variability, or in layman's terms, the differences that might arise simply because we are surveying a sample and not the whole population.

Confidence Level

The confidence level is a statistical indication of how much faith we can have in the findings from a sample. It's designated in terms of percentage, often seen at 95%, 99%, or 90%, indicating the likelihood that the sample result represents the true population parameter.

• A 95% confidence level implies that if a survey were conducted 100 different times, you'd expect the findings to reflect the true population parameter 95 times out of 100.
• Higher confidence levels indicate a higher chance of reflecting true population figures. However, they also result in wider confidence intervals, meaning more margin of error.

The choice of confidence level can affect the required sample size for your survey. To maintain accuracy and a lower margin of error at high confidence levels, a larger sample size is typically necessary.

Population Proportion

The population proportion is what we aim to estimate in situations like the president's policy support survey. It represents the fraction of the entire population that possesses a particular characteristic or opinion.

Knowing or having an estimate of the population proportion can greatly aid sample size calculations. For instance, if a previous survey indicates that 60% support a policy, this can be used to guide decisions on how many people to include in a new survey to ensure similar accuracy.

If no estimate is available, using 50% as a proxy is common. The assumption is based on 50% representation providing maximum variability, and hence the largest necessary sample, ensuring the survey's findings would still be generalizable across the population.

Critical Value

In the context of sample size estimation, the critical value is tied to the confidence level chosen for the survey. It is a point on the standard normal distribution that corresponds to the confidence level margin.

This value can be obtained from a Z-table or calculated using statistical software. For a 95% confidence level, the critical value is 1.96, which is commonly used in many sample size calculations.

• The critical value is an essential component of the formula used to calculate how large a sample needs to be, considering both the margin of error and the estimated population proportion.
• It directly influences the width of the confidence interval: a higher critical value means a wider interval, accommodating more potential variability in the sample's outcomes.

Understanding the role of the critical value helps to grasp how statistical accuracy and sample size are connected, ensuring survey results can be confidently projected to the broader population.