Question

derek earned pocket money by recycling cans and bottles on 7 occasions. the redemption values were: $9.00 $4.00 $1.00 $9.00 $4.00 $7.00 $1.00 what was the standard deviation of the redemption values he received? if the answer is a decimal, round it to the nearest ten cents. standard deviation (σ): $

Explanation:

Step1: Calculate the mean

The formula for the mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\).
Here \(n = 7\), \(x_1=9\), \(x_2 = 4\), \(x_3=1\), \(x_4 = 9\), \(x_5=4\), \(x_6 = 7\), \(x_7=1\).
\(\sum_{i=1}^{7}x_{i}=9 + 4+1+9+4+7+1=35\)
\(\bar{x}=\frac{35}{7}=5\)

Step2: Calculate the variance

The formula for the variance \(\sigma^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n}\)
\((x_1-\bar{x})^{2}=(9 - 5)^{2}=16\)
\((x_2-\bar{x})^{2}=(4 - 5)^{2}=1\)
\((x_3-\bar{x})^{2}=(1 - 5)^{2}=16\)
\((x_4-\bar{x})^{2}=(9 - 5)^{2}=16\)
\((x_5-\bar{x})^{2}=(4 - 5)^{2}=1\)
\((x_6-\bar{x})^{2}=(7 - 5)^{2}=4\)
\((x_7-\bar{x})^{2}=(1 - 5)^{2}=16\)
\(\sum_{i = 1}^{7}(x_{i}-\bar{x})^{2}=16+1+16+16+1+4+16=70\)
\(\sigma^{2}=\frac{70}{7}=10\)

Step3: Calculate the standard deviation

The formula for the standard deviation \(\sigma=\sqrt{\sigma^{2}}\)
Since \(\sigma^{2}=10\), \(\sigma=\sqrt{10}\approx 3.2\)

Answer:

\(3.2\)