Question

listed below in order are prices in dollars for a big mac hamburger in the united states, canada, mexico, china, japan, russia, switzerland, italy, spain, britain, india, and egypt. such data are used to compare currency exchange rates and the costs of goods in different countries. find the range, variance, and standard deviation for the given sample data. what do the measures of variation tell us about the prices of a big mac in different countries? 5.26 5.30 2.61 3.19 3.40 2.32 6.78 5.09 4.81 4.37 2.80 1.85 the range is 4.93 dollars. (type an integer or decimal rounded to two decimal places as needed.) the variance is (type an integer or vo decimal places as needed) dollars^2 dollars

Explanation:

Step1: Calculate the mean

The sample data is \(x = [5.26,5.30,2.61,3.19,3.40,2.32,6.78,5.09,4.81,4.37,2.80,1.85]\).
The formula for the mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\), where \(n = 12\).
\(\sum_{i=1}^{12}x_{i}=5.26 + 5.30+2.61+3.19+3.40+2.32+6.78+5.09+4.81+4.37+2.80+1.85=47.78\)
\(\bar{x}=\frac{47.78}{12}\approx3.9817\)

Step2: Calculate the variance

The formula for the sample variance \(s^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}\)
\((x_{1}-\bar{x})^{2}=(5.26 - 3.9817)^{2}\approx1.634\)
\((x_{2}-\bar{x})^{2}=(5.30 - 3.9817)^{2}\approx1.738\)
\((x_{3}-\bar{x})^{2}=(2.61 - 3.9817)^{2}\approx1.881\)
\((x_{4}-\bar{x})^{2}=(3.19 - 3.9817)^{2}\approx0.627\)
\((x_{5}-\bar{x})^{2}=(3.40 - 3.9817)^{2}\approx0.339\)
\((x_{6}-\bar{x})^{2}=(2.32 - 3.9817)^{2}\approx2.761\)
\((x_{7}-\bar{x})^{2}=(6.78 - 3.9817)^{2}\approx7.831\)
\((x_{8}-\bar{x})^{2}=(5.09 - 3.9817)^{2}\approx1.228\)
\((x_{9}-\bar{x})^{2}=(4.81 - 3.9817)^{2}\approx0.686\)
\((x_{10}-\bar{x})^{2}=(4.37 - 3.9817)^{2}\approx0.151\)
\((x_{11}-\bar{x})^{2}=(2.80 - 3.9817)^{2}\approx1.396\)
\((x_{12}-\bar{x})^{2}=(1.85 - 3.9817)^{2}\approx4.544\)

\(\sum_{i = 1}^{12}(x_{i}-\bar{x})^{2}=1.634+1.738 + 1.881+0.627+0.339+2.761+7.831+1.228+0.686+0.151+1.396+4.544 = 24.816\)

\(s^{2}=\frac{24.816}{12 - 1}=\frac{24.816}{11}\approx2.26\)

Answer:

The variance is \(2.26\) dollars\(^{2}\)