Problem 21 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

A study of the records of 85,000 apartment units in the greater Boston area revealed the following data: $$\begin{array}{llllll}\hline \text { Year } & 2002 & 2003 & 2004 & 2005 & 2006 \\ \hline \text { Occupancy } & & & & & \\\\\text { Rate, \% } & 95.6 & 94.7 & 95.2 & 95.1 & 96.1 \\\\\hline\end{array}$$ Find the average occupancy rate for the 5 yr in question. What is the standard deviation for these data?

Step-by-Step Solution

Answer

The average occupancy rate for the 5 years in question is \(95.34\%\) and the standard deviation is approximately \(0.4758\%\).

Step 1: Calculate the Mean

To find the mean (average) occupancy rate, we will sum up the occupancy rates for all the given years and then divide by the number of years. Mean (M) = \(\frac{(95.6+94.7+95.2+95.1+96.1)}{5}\)

Step 2: Simplify and find the Mean

M = \(\frac{(476.7)}{5}\) M = \(95.34\) The average (mean) occupancy rate for the years in question is 95.34%.

Step 3: Calculate the Variance (σ²)

To find the variance, we will first calculate the differences of each data point from the mean, square these differences, and then take the average of these squared differences. Variance (σ²) = \(\frac{(\text{sum of squared differences})}{\text{number of data points}}\) Squared Differences: 1. \((95.6 - 95.34)^2 = 0.0676\) 2. \((94.7 - 95.34)^2 = 0.4096\) 3. \((95.2 - 95.34)^2 = 0.0196\) 4. \((95.1 - 95.34)^2 = 0.0576\) 5. \((96.1 - 95.34)^2 = 0.5776\) Sum of squared differences = 1.132 Variance, σ² = \(\frac{1.132}{5}\)

Step 4: Simplify and find the Variance

σ² = \(0.2264\) The variance (σ²) = 0.2264.

Step 5: Calculate the Standard Deviation (σ)

To find the standard deviation, we simply take the square root of the variance. Standard Deviation (σ) = \(\sqrt{(0.2264)}\)

Step 6: Simplify and find the Standard Deviation

σ = \(0.4758\) The standard deviation for these data is approximately 0.4758% So, the average occupancy rate for the 5 years in question is \(95.34\%\) and the standard deviation is approximately \(0.4758\%\).

Key Concepts

Mean Calculation

The mean, often referred to as the average, is a measure of the central tendency of a dataset. It is calculated by adding up all the values in a dataset and then dividing by the number of values. In the context of occupancy rate statistics, the mean occupancy rate gives us a single number that represents the typical occupancy across multiple time periods.

In this exercise, to calculate the average occupancy rate for the apartment units over the span of five years, we added the annual occupancy rates together and then divided by five, since data was available for five years. Simplifying this further, we obtained the mean occupancy rate as 95.34%. This value is a simple and foundational statistic that can be used to compare the occupancy performance over time or with other datasets.

Variance Calculation

Variance is a statistical measure that tells us how spread out the numbers are in a dataset. It is the average squared differences from the Mean. A small variance indicates that the numbers are close to the mean and to each other, while a larger variance means that the numbers are more spread out.

In our example, we first determined each year's deviation from the mean occupancy rate and then squared these deviations to avoid negative values impacting the measure. After obtaining the squared differences for each year, we summed them to get the total squared difference. This total was then divided by the number of years to obtain the variance, which was calculated to be 0.2264. This indicates the variability of the annual occupancy rates around the average occupancy rate.

Standard Deviation Calculation

Standard deviation is a widely used measure of variability or diversity used in statistics and probability theory. It shows how much variation exists from the average (mean). A low standard deviation means that the data points tend to be close to the mean, whereas a high standard deviation indicates that the data points are spread out over a wider range of values.

To calculate the standard deviation for the occupancy rates, we took the square root of the variance. This step reverses the squaring process for variance, bringing the measure back to the original unit of measure and thus making it interpretable and comparable to the mean. The result was a standard deviation of approximately 0.4758%, implying that the annual occupancy rates do not stray very far from the mean rate, signifying a rather stable occupancy rate over the five year period in the greater Boston area.