Problem 49 - Basic Statistics for Business & Economics

Question

You want to estimate the mean family income in a rural area of central Indiana. The question is, how many families should be sampled? In a pilot sample of 10 families, the standard deviation of the sample was 500 . The sponsor of the survey wants you to use the \(95 \%\) confidence level. The estimate is to be within 100 How many families should be interviewed?

Step-by-Step Solution

Answer

97 families should be interviewed.

Step 1: Determine the Required Formula

To find the sample size needed, we use the formula for estimating sample size based on margin of error: \[ n = \left(\frac{Z \cdot \sigma}{E}\right)^2 \]Where:- \(n\) is the sample size.- \(Z\) is the Z-value for the specified confidence level, which is 1.96 for a 95% confidence level.- \(\sigma = 500\) is the sample standard deviation.- \(E = 100\) is the margin of error.

Step 2: Plug in the Known Values

Using the values provided:- \(Z = 1.96\)- \(\sigma = 500\)- \(E = 100\)We substitute them into the formula:\[ n = \left(\frac{1.96 \times 500}{100}\right)^2 \]

Step 3: Calculate the Sample Size

Now calculate the expression within the formula:\[ n = \left(\frac{980}{100}\right)^2 = (9.8)^2 = 96.04 \]Since the sample size must be a whole number, we round up to the nearest whole number: 97.

Key Concepts

Confidence Interval

Confidence intervals are a fundamental concept in statistics used to estimate the range within which a population parameter lies. When you hear the term "confidence interval," think of it as a "range of certainty." This range gives statisticians a measure of how sure they can be about their estimates from sample data.

For instance, if you conduct a survey or experiment, the exact population parameter (like the mean income) isn't always clear. A confidence interval provides a range, calculated from your sample data, that likely contains the true population parameter. Commonly, people use a 95% confidence level, which means that if you were to take 100 different samples and calculate the interval each time, about 95 of those intervals would contain the true population mean.

• The confidence interval depends on the variability in your data; less variability gives a narrower interval.
• A higher confidence level means a wider interval, as you're more certain the range covers the true parameter.

Understanding confidence intervals helps to better grasp how well your sample data represents the larger population.

Margin of Error

The margin of error is a critical statistic that describes the amount of random sampling error in a survey's results. Essentially, it's the "wiggle room" you allow in your estimates. For example, if the margin of error is 100 in a survey estimating mean family income, this suggests that the true mean could be 100 units higher or lower than the sample mean.

Margin of error is essential because it quantifies how precise or "accurate" your estimates might be. Narrower margins suggest more precise estimates, while a wider margin indicates more uncertainty in the estimates. A smaller margin is preferred when high precision is necessary, such as in policy-making or financial forecasts.

• Margin of error is inversely related to sample size: larger samples tend to reduce the margin of error.
• Adjusting the margin of error can affect the sample size needed for your study, requiring strategic planning and resource allocation.

Knowing the margin of error can help assess the reliability of survey and research results.

Standard Deviation

Standard deviation is a measure of how spread out the values in a data set are, or how much the individual data points deviate from the mean. In simpler terms, it tells us how much variation there is from the "average" value.

If the standard deviation is large, it means the values are spread out over a wider range. Conversely, a small standard deviation indicates that the values are clustered closely around the mean. In the context of determining sample sizes, as with the exercise provided, the standard deviation plays a crucial role.

• The larger the standard deviation, the larger the sample size needed to achieve a specific margin of error.
• It is a foundational element in many statistical formulae, including those for confidence intervals and margin of error.

Having a good estimate of standard deviation allows you to better design your study and optimize resources for more accurate conclusions.