Problem 2 - Basic Statistics for Business and Economics with Student

Question

A sample of 81 observations is taken from a normal population with a standard deviation of \(5 .\) The sample mean is \(40 .\) Determine the 95 percent confidence interval for the population mean.

Step-by-Step Solution

Answer

The 95% confidence interval for the population mean is (38.91, 41.09).

Step 1: Identify the Known Values

From the problem, we know the sample size \( n = 81 \), the sample standard deviation \( \sigma = 5 \), and the sample mean \( \bar{x} = 40 \).

Step 2: Determine the Z-score for 95% Confidence

A 95% confidence interval corresponds to the Z-score that captures 95% of the data from a standard normal distribution. For a two-tailed test, this Z-score is \( z = 1.96 \).

Step 3: Calculate the Standard Error

The standard error (SE) of the sample mean is calculated using the formula \( SE = \frac{\sigma}{\sqrt{n}} \). Substituting the known values gives \( SE = \frac{5}{\sqrt{81}} = \frac{5}{9} \approx 0.556 \).

Step 4: Compute the Margin of Error

The margin of error (ME) is found by multiplying the Z-score by the standard error. So, \( ME = 1.96 \times 0.556 \approx 1.0896 \).

Step 5: Determine the Confidence Interval

The 95% confidence interval is calculated using the formula \( \bar{x} \pm ME \). Here, it becomes \( 40 \pm 1.0896 \). Simplifying this, the interval is \( (38.9104, 41.0896) \).

Key Concepts

Sample Mean

The sample mean is a crucial concept in statistics and represents the average value of a set of observations. It is denoted as \( \bar{x} \), and it is calculated by summing all the observations in a sample and then dividing by the total number of observations in that sample.
The formula for calculating the sample mean is:\[\bar{x} = \frac{\sum{x_i}}{n}\]Where:

• \( \sum{x_i} \) is the sum of all observed values.
• \( n \) is the number of observations.

In the context of the exercise, the sample mean \( \bar{x} = 40 \) is calculated from 81 observations. The sample mean is often used in inferential statistics to estimate the population mean, especially when direct measurement of the population is impractical.

Standard Error

The concept of standard error (SE) is fundamental in understanding how the sample mean represents the population mean. It provides an estimate of the variability of the sample mean from the population mean.

The formula for calculating the standard error is:\[SE = \frac{\sigma}{\sqrt{n}}\]Where:

• \( \sigma \) is the standard deviation of the population.
• \( n \) is the number of observations in the sample.

In the exercise example, the standard deviation \( \sigma \) is given as \( 5 \), and the standard error is calculated using the sample size of 81, resulting in an SE of approximately \( 0.556 \). This small standard error indicates that there is little variability between the sample mean and the true population mean, suggesting a reliable sample.

Z-score

The Z-score ties the observed data to the standard normal distribution, allowing statisticians to understand probabilities and make inferences about the population. In this context, the Z-score is crucial for calculating the confidence interval.
A 95% confidence interval means that we expect 95% of sample means to lie within this interval when repeated samples are taken. For a 95% confidence level in a standard normal distribution, the Z-score is commonly \( 1.96 \).
This Z-score reflects the number of standard deviations an observation is from the mean. By using the Z-score, we can scale the standard error to capture the desired level of confidence around the sample mean.

Margin of Error

The margin of error is an essential component of confidence intervals, quantifying the range within which we expect the true population mean to lie, relative to the sample mean.
The margin of error is calculated by multiplying the Z-score by the standard error:\[ME = Z \times SE\]In our exercise, the margin of error is computed as \( 1.96 \times 0.556 \approx 1.0896 \).
This value is then used to establish the range of the confidence interval around the sample mean. For example, with a sample mean of 40, the confidence interval would fall between \( 38.9104 \) and \( 41.0896 \), indicating that we are 95% confident that the true population mean is within this range.