Problem 23 - Basic Statistics for Business and Economics with Student

Question

A population is estimated to have a standard deviation of \(10 .\) We want to estimate the population mean within 2, with a 95 percent level of confidence. How large a sample is required?

Step-by-Step Solution

Answer

The required sample size is 97.

Step 1: Understanding the Requirements

We need to find how large a sample is required to estimate the population mean within a certain margin of error using the formula for sample size in confidence intervals. The given information includes the population standard deviation, desired margin of error, and confidence level.

Step 2: Identifying the Formula

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

Step 3: Finding the Z-score

For a 95% confidence level, the Z-score is approximately 1.96. This Z-score corresponds to the standard normal distribution.

Step 4: Substituting Values into the Formula

Using the values given in the problem: \(Z = 1.96\), \(\sigma = 10\), and \(E = 2\), substitute these into the sample size formula. \[n = \left( \frac{1.96 \cdot 10}{2} \right)^2 \]

Step 5: Calculating the Sample Size

Perform the calculation as follows: Calculate \( 1.96 \times 10 = 19.6 \). Then divide by the margin of error (2): \( 19.6 / 2 = 9.8 \). Square this result to find \(n\):\[ n = 9.8^2 = 96.04 \]

Step 6: Rounding the Sample Size

Since the sample size must be a whole number, round up to ensure the margin of error requirement is met. Thus, \( n = 97 \).

Key Concepts

Confidence Interval

A confidence interval is a range of values we are fairly certain includes the true population parameter. In statistics, when we don't know the exact value of a population parameter, we estimate it with a confidence interval. The range is determined by the data from a sample and is expressed with a certain level of confidence, like 95% or 99%.
For a 95% confidence interval, you can imagine it as saying we are 95% confident that the interval we've calculated contains the true mean of the population. This does not imply that 95% of the population values are within this range, but that if we repeatedly took samples and calculated confidence intervals, approximately 95% of them would contain the true population mean.
Calculating a confidence interval involves the sample mean, the population standard deviation, and the Z-score corresponding to the desired confidence level. These elements help define the width and reliability of the confidence interval.

Margin of Error

The margin of error represents how much error is acceptable when estimating the population mean. It is the range above and below the sample statistic—for instance, a sample mean—within which we expect the population parameter to fall.
In the formula \( n = \left( \frac{Z \cdot \sigma}{E} \right)^2 \), \(E\) is the margin of error. It's how precise you want your estimation to be. A smaller margin of error requires a larger sample size to ensure higher accuracy in estimating the population parameter.
So, if you want to estimate the population mean within 2 units, your margin of error is 2. This means you're willing to accept a sample mean that equals the population parameter plus or minus 2.

Population Standard Deviation

The population standard deviation, represented as \(\sigma\), is a measure of how much variation or dispersion exists within a population. A lower standard deviation indicates that values tend to be closer to the mean, whereas a higher standard deviation means values are spread out over a wider range.
Knowing the population standard deviation is crucial in calculating the sample size needed for a specific confidence interval and margin of error.
When we have a known population standard deviation, we can plug it into the sample size formula: \[ n = \left( \frac{Z \cdot \sigma}{E} \right)^2 \]
It directly impacts how many samples you need. A larger standard deviation often requires a larger sample size because the data is more spread out, necessitating more data to ensure that the sample mean is close to the population mean.

Z-score

The Z-score is an essential component when calculating confidence intervals and sample sizes. It quantifies how many standard deviations an element is from the mean. With a standard normal distribution, the Z-score will determine how confident you are about the location of the population mean.
For example, for a 95% confidence level, the Z-score is approximately 1.96. This just means that you are assuming that 95% of your data falls within 1.96 standard deviations of the mean in a normal distribution.
To find a Z-score, you can use statistical tables or software that provide the correct Z-value for a given confidence level. It's necessary because it standardizes our results, allowing comparisons across different contexts and datasets.
In sample size formulas, the Z-score ensures that the interval around your sample mean will capture the true population mean a certain percentage of the time, as defined by your confidence level.