Problem 53 - Basic Statistics for Business and Economics with Student

Question

You are to conduct a sample survey to determine the mean family income in a rural area of central Florida. 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 percent 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: Identify Known Values

We need to determine the sample size \( n \) required. From the problem, we know that the sample standard deviation \( s = 500 \), the desired margin of error \( E = 100 \), and the confidence level is 95%. The corresponding z-score for a 95% confidence level is approximately 1.96.

Step 2: Use Sample Size Formula

The formula to calculate the sample size for estimating a population mean is given by:\[ n = \left( \frac{z \cdot s}{E} \right)^2 \] where \( z \) is the z-score, \( s \) is the standard deviation, and \( E \) is the margin of error.

Step 3: Plug in Values

Plug the known values into the formula:\[ n = \left( \frac{1.96 \times 500}{100} \right)^2 \]

Step 4: Calculate the Expression

First, calculate the expression inside the parentheses:\[ \frac{1.96 \times 500}{100} = \frac{980}{100} = 9.8 \]

Step 5: Square the Result

Now, square the result of step 4 to find the sample size:\[ n = (9.8)^2 = 96.04 \]

Step 6: Round Up to Nearest Whole Number

Since the sample size must be a whole number, round 96.04 up to the nearest whole number, which is 97.

Key Concepts

Confidence Interval

When we talk about confidence intervals, we're discussing a range within which we expect a population parameter to lie, based on our sample data. The idea is to provide an estimate that we can be fairly confident about. For example, in social science surveys, you might often hear phrases like "with a 95% certainty, the results support...". In mathematical terms, a 95% confidence interval suggests that if you repeated a study 100 times, in 95 of those times, the true population parameter would fall within this interval.

A confidence interval consists of everything in between the upper and lower bounds. The larger the interval, the less precise our estimate becomes, whereas a smaller interval indicates more precision. The width of this interval is largely determined by the confidence level (like 95% or 99%), the sample size, and variability within the data.

Margin of Error

The margin of error is simply the "plus or minus" figure often quoted in survey results or statistical findings. It tells us how much we can expect our sample result to vary from the true population value. It's crucial for understanding the reliability of polling or survey data.

The margin of error is directly related to the confidence level and the variability in the sample. A smaller margin of error means more confidence that your results accurately reflect the population. To calculate this, we use the formula: \( E = z \cdot \frac{s}{\sqrt{n}} \) where \( z \) is the z-score, \( s \) is the standard deviation, and \( n \) is the sample size.

An important point to remember when considering margin of error is that a smaller sample size means a larger margin of error, implying less confidence in the results.

Standard Deviation

Standard deviation is a statistical measure that quantifies the amount of variation or dispersion in a set of data. In simple terms, it tells us how much individual data points in a sample deviate from the mean or average value.

This is very important because it can influence the sample size needed for a survey or study. Larger standard deviations indicate more variability within the data set, which requires a larger sample to effectively estimate the population mean. Conversely, a smaller standard deviation suggests less variability, often requiring a smaller sample size.

In our exercise, a standard deviation of \( \\(500 \) means that on average, each family's income is about \( \\)500 \) away from the mean income of the group surveyed.

Z-score

A z-score is a statistical measurement that describes a value's relation to the mean of a group of values, measured in units of standard deviation. If you have a z-score of 0, it means your data point is exactly average, whereas a z-score of +1 or -1 indicates one standard deviation above or below the mean, respectively.

In the context of confidence intervals, the z-score is used to express the desired level of certainty. For example, a 95% confidence level corresponds to a z-score of approximately 1.96. This tells us that we're almost two standard deviations away from the mean, which covers about 95% of data in a normal distribution.

Using z-scores to determine sample size, as shown in the exercise, allows researchers to ensure the precision and reliability of their estimations.

Population Mean Estimation

Estimating the population mean is a common goal in statistics, especially when it is impractical to measure an entire population. Instead, researchers use a sample and apply statistical methods to make educated guesses about the population mean.

In our example, the goal was to estimate the average family income in a rural area of central Florida. By determining an adequate sample size using the formula \( n = \left( \frac{z \cdot s}{E} \right)^2 \), we can make that estimation with a specified level of confidence and accuracy.

This process involves understanding the variation within the data (via standard deviation) and the desired precision (using margin of error). With this estimation method, we gain insights into the broader population behaviors or characteristics with manageable resources.