Problem 29 - Basic Statistics for Business & Economics

Question

As part of their business promotional package, the Milwaukee Chamber of Commerce would like an estimate of the mean cost per month to lease a one- bedroom apartment. The mean cost per month for a random sample of 40 apartments currently available for lease was 884. The standard deviation of the sample was 50. a. Develop a 98% confidence interval for the population mean. b. Would it be reasonable to conclude that the population mean is 950 per month?

Step-by-Step Solution

Answer

a. The 98% confidence interval is (865.59, 902.41). b. No, 950 is outside the interval.

Step 1: Understand the Scenario

We are given a random sample of 40 one-bedroom apartments with an average (mean) monthly cost of $884 and a standard deviation of $50. We need to estimate the range in which the true mean cost of all one-bedroom apartments lies with 98% confidence.

Step 2: Identify the Formula for Confidence Interval

The formula for finding a confidence interval for the mean when the population standard deviation is unknown is \( \bar{x} \pm z_{\alpha/2} \frac{s}{\sqrt{n}} \), where \( \bar{x} \) is the sample mean, \( s \) is the sample standard deviation, \( n \) is the sample size, and \( z_{\alpha/2} \) is the critical value from the standard normal distribution for the desired confidence level.

Step 3: Find the Critical Value

For a 98% confidence interval, the critical value \( z_{\alpha/2} \) can be found using a standard normal distribution table or calculator. It is approximately 2.33.

Step 4: Calculate Margin of Error

The margin of error can be calculated as \( z_{\alpha/2} \frac{s}{\sqrt{n}} = 2.33 \times \frac{50}{\sqrt{40}} = 18.41 \).

Step 5: Calculate the Confidence Interval

Apply the formula: \( \bar{x} \pm \text{Margin of Error} = 884 \pm 18.41 \), which results in the interval \( (865.59, 902.41) \).

Step 6: Interpret the Confidence Interval and Answer Part b

The confidence interval for the population mean is \( (865.59, 902.41) \) with 98% confidence. Since 950 is not within this interval, it would not be reasonable to conclude that the true population mean cost is 950 per month.

Key Concepts

Population Mean

In statistics, the population mean is denoted as \( \mu \). It represents the average of a set of values across an entire population. Understanding the population mean helps give context when comparing sample data. When we talk about a population mean, we're speaking of a true average, although the exact value is often unknown. That's why we estimate it using a sample mean. The example given involves the mean cost per month to lease a one-bedroom apartment.The sample mean, denoted as \( \bar{x} \), is taken from a random sample and serves as an estimate of the population mean. By examining the mean cost from our sample of apartments, we use this sample mean and look to predict the actual mean for the entire population of available apartments.

Sample Standard Deviation

The standard deviation is a measure of how spread out numbers are in a set. More specifically, the sample standard deviation, represented as \( s \), indicates the extent of variation in our sample data. Knowing this allows us to understand how much variation or "spread" is present compared to our sample mean.For instance, in the given exercise, we have a sample standard deviation of 50. This shows us that the costs in our sample don't deviate wildly from one another by much -- on average, they vary by $50 from the mean. Larger standard deviations suggest greater variability in the data, and smaller ones indicate less spread.Employing sample standard deviation helps when calculating the confidence interval, giving us an idea of the true variability in the population when the population standard deviation is unknown.

Critical Value

In the context of confidence intervals, the critical value is crucial because it sets the bounds of our estimate's reliability. For a confidence interval, this value, often denoted as \( z_{\alpha/2} \), is taken from the statistical distribution that applies to our data.The critical value is chosen based on our desired level of confidence. In the example, a 98% confidence level requires a critical value of approximately 2.33 from the standard normal distribution. This indicates that we're using 98% of the central values from a normal distribution to establish our confidence interval.Selecting the right critical value is essential for accurately estimating the range in which we expect the population mean to fall with our selected certainty level.