Problem 20 - Basic Statistics for Business & Economics

Question

We want to estimate the population mean within \(5,\) with a \(99 \%\) level of confidence. The population standard deviation is estimated to be \(15 .\) How large a sample is required?

Step-by-Step Solution

Answer

A sample size of 60 is required.

Step 1: Understanding the Problem

We need to determine the sample size needed to estimate the population mean with a margin of error of 5, given a confidence level of 99% and a population standard deviation of 15.

Step 2: Recall the Formula for Sample Size

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

Step 3: Identify the Components

We are given:- \( \sigma = 15 \)- \( E = 5 \)- Since we need a 99% confidence level, we use a Z-score of approximately 2.576 (from Z-tables).

Step 4: Plug Values into the Formula

Substitute the known values into the sample size formula:\[ n = \left( \frac{2.576 \cdot 15}{5} \right)^2 \]

Step 5: Calculate the Sample Size

First, multiply 2.576 by 15:\[ 2.576 \times 15 = 38.64 \]Next, divide by the margin of error 5:\[ \frac{38.64}{5} = 7.728 \]Finally, square the result to find the sample size:\[ 7.728^2 \approx 59.73 \]

Step 6: Round Up the Sample Size

Since we can't have a fraction of a sample, round up to the nearest whole number: Hence, the required sample size is 60.

Key Concepts

Confidence Interval

The concept of a confidence interval is key in statistics. It represents a range in which we expect the population parameter, like a mean or proportion, to lie with a certain level of confidence. For example, in the problem, we aim for a 99% confidence interval, meaning that we are 99% sure the true population mean falls within this interval.
To determine this range, two critical components are necessary: the sample statistic, like a sample mean, and the margin of error, which accounts for the uncertainty inherent in sampling. The interval is calculated by starting with the sample statistic and adjusting it by adding and subtracting the margin of error. Thus, it takes the form:

• Lower limit = Sample mean - Margin of error
• Upper limit = Sample mean + Margin of error

To widen or narrow this interval depends on the chosen confidence level, the variability of the data, and the size of the sample. Increasing the confidence level or decreasing the margin of error typically results in a wider interval.

Margin of Error

The margin of error quantifies the degree of uncertainty or possible error in our sample results. It's a critical measure that indicates how much the sample mean could differ from the actual population mean. In the exercise, a margin of error of 5 was selected, meaning that our sample estimate should be within 5 units of the true population mean.
This margin is calculated based on the standard deviation and the sample size, often incorporating the Z-score for the desired confidence level. The formula:

• Margin of error (E) = Z \( \cdot \) \( \sigma \) / \( \sqrt{n} \)

The margin of error decreases as the sample size increases. Therefore, a larger sample size will provide a more accurate estimate of the population mean. Adjusting the margin of error can balance the need for precision against practical considerations such as time, cost, and resources.

Population Standard Deviation

Population standard deviation is a key statistical measure that tells us the extent of variation or dispersion of a set of data points in a population. It measures how much the individual data points deviate from the population mean. In our exercise, the standard deviation is provided as 15, which helps us understand the spread of the data around the mean.
A higher standard deviation means the data points are spread out over a wider range of values, and a lower standard deviation indicates that they are clustered more closely around the mean. This measure is crucial as it factors into the calculation of both the margin of error and the confidence interval. The larger the standard deviation, the bigger the margin of error becomes, leading to a wider confidence interval. Understanding this concept helps to grasp how variability within the data affects the reliability and accuracy of our statistical estimations.

Z-score

The Z-score is a statistical measurement that describes a value's relation to the mean of a group of values, expressed in terms of standard deviations. It helps determine how far away a particular value is from the mean, either above or below it. In the context of our sample size calculation, the Z-score is linked to the confidence level.
For a 99% confidence level used in the problem, the Z-score is approximately 2.576. This value is derived from standard Z-tables, which convert the confidence level into a critical value used for calculations. It effectively scales the population standard deviation to determine the margin of error, giving us control over the accuracy of our estimates.

• Z-score increases with higher confidence levels, showing more variability included.
• A higher Z-score leads to a wider confidence interval.
• It ensures that, with 99% certainty, the true mean lies within the calculated interval.

Understanding Z-scores is essential for accurate sample size calculations and forming confidence intervals.