Problem 42 - Basic Statistics for Business and Economics with Student

Question

An important factor in selling a residential property is the number of people who look through the home. A sample of 15 homes recently sold in the Buffalo, New York, area revealed the mean number looking through each home was 24 and the standard deviation of the sample was 5 people. Develop a 98 percent confidence interval for the population mean.

Step-by-Step Solution

Answer

The 98% confidence interval is (20.16, 27.84).

Step 1: Identify Relevant Parameters

You have a sample of 15 homes, where the sample mean \( \bar{x} = 24 \) and the sample standard deviation \( s = 5 \). The sample size is \( n = 15 \), and we aim to develop a 98% confidence interval for the population mean.

Step 2: Determine the Critical Value

Since the sample size is small and less than 30, the T-distribution will be used. For a 98% confidence interval with \( n-1 = 14 \) degrees of freedom, locate the critical value \( t^* \) using a T-table or technology. The value of \( t^* \) for 98% confidence level and 14 degrees of freedom is approximately 2.977.

Step 3: Calculate the Margin of Error

The margin of error (ME) is calculated using the formula: \[ ME = t^* \times \frac{s}{\sqrt{n}} \]Substitute the known values:\[ ME = 2.977 \times \frac{5}{\sqrt{15}} \approx 3.84 \]

Step 4: Construct the Confidence Interval

The confidence interval for the population mean is given by:\[ \bar{x} \pm ME \]Using the sample mean and the margin of error:\[ 24 \pm 3.84 \]This results in a confidence interval:\( (20.16, 27.84) \).

Step 5: Interpret the Confidence Interval

The 98% confidence interval for the population mean number of people looking through these homes is between 20.16 and 27.84. This means we are 98% confident that the true population mean lies in this range.

Key Concepts

Sample Mean

The concept of the sample mean is fundamental in statistics and helps in understanding general tendencies in the data. In this specific context, the sample mean represents the average number of people who looked through the homes in the sample. It is calculated by summing up all the observed values (number of visitors for each house) and dividing by the total number of observations.

• The formula is expressed as: \( \bar{x} = \frac{\sum x_i}{n} \), where \( x_i \) is each individual observation and \( n \) is the total number of observations.

In our exercise, the sample mean is 24, meaning that on average, 24 people looked through each home. Understanding this average helps us make inferences about the larger population. Breaking down the data to an average makes it manageable and simplifies the analysis.
By using the sample mean, statisticians can draw conclusions about the population mean, which is crucial for developing confidence intervals.

Standard Deviation

Standard deviation is a measure of how spread out the numbers in a data set are. It gives insight into the variability or diversity of the data. In simpler terms, it tells you how much the individual data points differ from the sample mean.

• The formula for the sample standard deviation is \( s = \sqrt{ \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2 } \), where \( x_i \) is each data point, \( \bar{x} \) is the sample mean, and \( n \) is the number of observations.

In this exercise, the standard deviation is given as 5, indicating the average spread of the number of visitors from the mean is 5. This measure allows us to understand the extent of variability in our data, which is crucial when we further apply it to calculate the margin of error for the confidence interval.
Standard deviation is an important component in assessing the reliability of the sample mean as a representation of the true population mean.

T-Distribution

When estimating population parameters from a small sample size, the T-distribution is preferable over the normal distribution. This is especially the case when the sample size is less than 30, as in our exercise where it is 15.

• The T-distribution resembles the normal distribution but has fatter tails, accounting for this additional variability in smaller samples.
• The key parameter for the T-distribution is degrees of freedom, calculated as \( n-1 \), where \( n \) is the number of observations.

In our example, the degrees of freedom is 14. Understanding the T-distribution is essential because it adjusts our margin of error and the resulting confidence interval. We use a T-distribution table or statistical software to find the critical value based on the desired confidence level and degrees of freedom.
Using the T-distribution ensures that the confidence interval is accurate for the population mean estimate, given the sample size.

Critical Value

The critical value is a cutoff point that defines the range for the confidence interval. For the T-distribution, it is represented by \( t^* \). The critical value depends on the confidence level and the degrees of freedom.

• In this context, it is used to determine how many standard deviations away from the sample mean we should go to capture the desired confidence level.

To find \( t^* \), we use a T-distribution table or software, focusing on the confidence level of 98% and the degrees of freedom calculated previously. The critical value is approximately 2.977.
Once we have the critical value, it is used to compute the margin of error with the formula \( ME = t^* \times \frac{s}{\sqrt{n}} \). This step is crucial as it defines the width of the confidence interval. Knowing the critical value brings us closer to making an informed and confident estimate about the population mean.