A supervisor finds the mean number of miles that the employees in a department live from work. He finds [tex]$\bar{x}=29$[/tex] and [tex]$s=3.6$[/tex]. Which statement must be true?
A. [tex]$z_{37}$[/tex] is within 1 standard deviation of the mean.
B. [tex]$z_{37}$[/tex] is between 1 and 2 standard deviations of the mean.
C. [tex]$z_{37}$[/tex] is between 2 and 3 standard deviations of the mean.
D. [tex]$z_{37}$[/tex] is more than 3 standard deviations of the mean.
Answer (2)
Calculate the z-score: z = 3.6 37 − 29 .
Simplify the expression: z = 3.6 8 .
Calculate the value: z ≈ 2.22 .
z 37 is between 2 and 3 standard deviations of the mean: z 37 is between 2 and 3 standard deviations of the mean.
Explanation
Understand the problem and provided data We are given that the mean number of miles employees live from work is x ˉ = 29 and the standard deviation is s = 3.6 . We want to determine how many standard deviations away from the mean the value z 37 = 37 is.
State the formula To find how many standard deviations z 37 is away from the mean, we use the formula: z = s x − x ˉ where x = 37 , x ˉ = 29 , and s = 3.6 .
Calculate the z-score Plugging in the values, we get: z = 3.6 37 − 29 = 3.6 8 Calculating this value: z = 3.6 8 ≈ 2.22
Determine the correct statement Since z ≈ 2.22 , z 37 is between 2 and 3 standard deviations of the mean.
Final Answer Therefore, the correct statement is: z 37 is between 2 and 3 standard deviations of the mean.
Examples
Understanding standard deviations helps in many real-world scenarios. For example, in quality control, if the average weight of a product is 100g with a standard deviation of 2g, knowing how many standard deviations a particular product's weight is from the mean helps determine if the product meets the required standards. Similarly, in finance, standard deviation is used to measure the volatility of an investment. In education, it can be used to understand the spread of scores around the average performance.
The z-score of z 37 = 37 is calculated to be approximately 2.22, which means it is between 2 and 3 standard deviations above the mean. Therefore, the correct option is C. z 37 is between 2 and 3 standard deviations of the mean.
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