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How do grocery costs compare across the country? Using a market basket of 10 items including meat, milk, bread, eggs, coffee, potatoes, cereal, and orange juice, Where to Retire magazine calculated the cost of the market basket in six cities and in six retirement areas across the country (Where to Retire, November/December 2003 ). The data with market basket cost to the nearest dollar are as follows: $$\begin{array}{lclr} \text { City } & \text { cost } & \text { Retirement Area } & \text { cost } \\\ \text { Buffalo, NY } & \$ 33 & \text { Biloxi-Gulfport, MS } & \$ 29 \\ \text { Des Moines, IA } & 27 & \text { Asheville, NC } & 32 \\ \text { Hartford, CT } & 32 & \text { Flagstaff, AZ } & 32 \\ \text { Los Angeles, CA } & 38 & \text { Hilton Head, SC } & 34 \\ \text { Miami, FL } & 36 & \text { Fort Myers, FL } & 34 \\ \text { Pittsburgh, PA } & 32 & \text { Santa Fe, NM } & 31\end{array}$$ a. Compute the mean, variance, and standard deviation for the sample of cities and the sample of retirement areas. b. What observations can be made based on the two samples?

Step by step solution

01

Calculate Mean for Cities

To calculate the mean cost for the sample of cities, sum all the market basket costs of the cities and divide by the number of cities. Costs for cities: 33 (Buffalo) + 27 (Des Moines) + 32 (Hartford) + 38 (Los Angeles) + 36 (Miami) + 32 (Pittsburgh) = 198. Number of cities = 6.Mean = \( \frac{198}{6} = 33 \).

02

Calculate Variance for Cities

To calculate the variance, find the difference between each city's cost and the mean, square it, sum all squared differences, and then divide by the number of cities minus 1. Squares of differences: \((33-33)^2 = 0\), \((27-33)^2 = 36\), \((32-33)^2 = 1\), \((38-33)^2 = 25\), \((36-33)^2 = 9\), \((32-33)^2 = 1\). Sum: 0 + 36 + 1 + 25 + 9 + 1 = 72.Variance: \( \frac{72}{5} = 14.4 \).

03

Calculate Standard Deviation for Cities

Standard deviation is the square root of the variance.\( \sqrt{14.4} \approx 3.79 \).

04

Calculate Mean for Retirement Areas

Sum the costs for the retirement areas and divide by the number of retirement areas. Costs for retirement areas: 29 (Biloxi-Gulfport) + 32 (Asheville) + 32 (Flagstaff) + 34 (Hilton Head) + 34 (Fort Myers) + 31 (Santa Fe) = 192.Number of retirement areas = 6.Mean = \( \frac{192}{6} = 32 \).

05

Calculate Variance for Retirement Areas

Calculate the variance similarly to cities. Find individual squared differences from the mean and sum them.Squares of differences: \((29-32)^2 = 9\), \((32-32)^2 = 0\), \((32-32)^2 = 0\), \((34-32)^2 = 4\), \((34-32)^2 = 4\), \((31-32)^2 = 1\). Sum: 9 + 0 + 0 + 4 + 4 + 1 = 18.Variance: \( \frac{18}{5} = 3.6 \).

06

Calculate Standard Deviation for Retirement Areas

Standard deviation is the square root of the variance.\( \sqrt{3.6} \approx 1.90 \).

07

Compare Observations

The mean cost is slightly higher in cities than in retirement areas. However, the variance and standard deviation are larger in cities, indicating more variability in grocery costs across cities compared to retirement areas.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mean Calculation

To find the average cost of a market basket in a group, you calculate the mean. This is a simple process of adding up all the values and dividing them by the number of items. For instance, when calculating the mean cost for cities, we sum the costs:

• Buffalo: \(33
• Des Moines: \)27
• Hartford: \(32
• Los Angeles: \)38
• Miami: \(36
• Pittsburgh: \)32

Summing these gives $198. Since there are six cities, we divide the total by six: \( \frac{198}{6} = 33 \) dollars. This number is the mean cost of the market basket for the cities.

The process is similar for retirement areas. Add up the costs and divide by six, resulting in a mean cost of \( \frac{192}{6} = 32 \) dollars for retirement areas. The mean provides a central value that gives us a basic sense of the typical cost.

Variance Calculation

Variance measures how much the costs differ from the mean. To calculate the variance, first estimate how each cost deviates from the mean. For example, in the case of cities:

• Calculate each difference from the mean. For Buffalo, the difference is \( (33 - 33) = 0 \).
• Square each difference: \( 0^2 = 0 \). Repeat for all costs.
• Add all squared differences: \( 0 + 36 + 1 + 25 + 9 + 1 = 72 \).

Finally, divide by one less than the number of cities: \( \frac{72}{5} = 14.4 \). This is the variance for city costs.

For retirement areas, perform the same steps. Calculate the differences, square them, sum up the squares, and divide by five, giving us a variance of \( \frac{18}{5} = 3.6 \). A higher variance means more spread out costs.

Standard Deviation Calculation

Standard deviation lets us know how spread out the costs are around the mean. It is simply the square root of the variance.

For the cities, the variance was \( 14.4 \), so the standard deviation is \( \sqrt{14.4} \approx 3.79 \). This tells us that cities have a wider range of costs.

For retirement areas, the variance was \( 3.6 \). Calculating the square root gives a smaller standard deviation: \( \sqrt{3.6} \approx 1.90 \). This implies more consistent costs compared to the cities.

In essence, standard deviation provides a tangible sense of variability, highlighting if costs tend to be similar or if they vary significantly.