Problem 12 - Basic Statistics for Business and Economics

Question

A recent article in the Cincinnati Enquirer reported that the mean labor cost to repair a heat pump is \(\$ 90\) with a standard deviation of \(\$ 22 .\) Monte's Plumbing and Heating Service completed repairs on two heat pumps this morning. The labor cost for the first was \(\$ 75\) and it was \(\$ 100\) for the second. Compute \(z\) values for each and comment on your findings.

Step-by-Step Solution

Answer

The first repair's z-value is -0.68 and the second's is 0.45; both costs are within one standard deviation from the mean.

Step 1: Understand the z-value formula

The z-value is calculated using the formula \( z = \frac{x - \mu}{\sigma} \), where \( x \) is the value of the sample, \( \mu \) is the mean of the population, and \( \sigma \) is the standard deviation of the population.

Step 2: Calculate the z-value for the first repair

For the first repair, \( x = 75 \). The mean (\( \mu \)) is 90 and the standard deviation (\( \sigma \)) is 22. Substitute these values into the formula: \[ z = \frac{75 - 90}{22} = \frac{-15}{22} \approx -0.68 \].

Step 3: Calculate the z-value for the second repair

For the second repair, \( x = 100 \). Substitute the values into the formula: \[ z = \frac{100 - 90}{22} = \frac{10}{22} \approx 0.45 \].

Step 4: Interpret the z-values

The z-value of \(-0.68\) for the first repair indicates that the cost was below the mean labor cost by 0.68 standard deviations. The z-value of \(0.45\) for the second repair indicates that the cost was above the mean by 0.45 standard deviations. These values show that both repairs had costs within roughly one standard deviation of the mean.

Key Concepts

Standard Deviation

Standard deviation is a fundamental concept in statistics that measures the amount of variation or dispersion in a set of values. It helps us understand how much the values in a data set deviate from the mean.
In simpler terms, it's a way to tell how spread out the numbers are from the average value. A larger standard deviation indicates that the data points are spread out over a wider range, while a smaller standard deviation signifies that they are clustered tightly around the mean.
Some key points about standard deviation are:

• Calculation: To calculate standard deviation, we first find the variance, which is the average of the squared differences from the mean. The square root of the variance gives us the standard deviation.
• Usefulness: It is used in finance, weather forecasting, and any field that deals with data, as it gives insight into the reliability and volatility of data.
• Units: The standard deviation is reported in the same units as the data, which makes it easier to interpret.

In the original exercise, a standard deviation of $22 signifies how much labor cost can differ from the average of $90 when repairing heat pumps. Both repair costs fall within one standard deviation, showcasing that the prices weren't overly unpredictable.

Mean in Statistics

The mean, often referred to as the average, is a central value that represents a set of data. To find the mean, you simply add up all the numbers and then divide by the number of values. It's a measure of central tendency that gives a quick snapshot of the overall tendency of a dataset.
There are several reasons why the mean is important:

• Indicator of central location: It provides a single value that represents the center of the data distribution, helping to summarize the information.
• Simple Calculation: Calculating the mean is straightforward and easy, even with large datasets.
• Predictive Use: It's often used as the basis for predicting future trends in data.

In the given problem, the mean labor cost of $90 acts as a benchmark. The z-scores derived from using the mean help identify how far or close each repair cost is from the average. This comparison is crucial in understanding cost consistency across different repairs.

Normal Distribution

Normal distribution, often referred to as the bell curve, is a probability distribution where most of the data points cluster around the mean, creating a bell-shaped curve. It is fundamental in statistics and is also known as a Gaussian distribution.
Some significant attributes of a normal distribution include:

• Symmetry: The graph is symmetric around the mean, which means the left side is a mirror image of the right side.
• Peak at Mean: The highest point on the curve is at the mean, indicating that most data points are close to the average value.
• Standard Deviations: The spread of the curve is defined by the standard deviation, with areas under the curve representing probabilities.

In the context of the exercise, assuming a normal distribution allows the use of z-scores to understand how far the repair costs deviate from the average cost of $90. The z-scores show how typical or atypical the repair costs are within the context of expected cost variations.