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?
[tex]$z_{37}$[/tex] is within 1 standard deviation of the mean.
[tex]$z_{37}$[/tex] is between 1 and 2 standard deviations of the mean.
[tex]$z_{37}$[/tex] is between 2 and 3 standard deviations of the mean.
[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 ≈ 2.22 .
Since 2 < 2.22 < 3 , z 37 is between 2 and 3 standard deviations from the mean.
The correct statement is: 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 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 z Plugging in the values, we get: z = 3.6 37 − 29 = 3.6 8 = 2.2222...
Determine the correct statement Since 2 < z < 3 , z 37 is between 2 and 3 standard deviations of the mean.
Final Answer Therefore, the statement that must be true 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, manufacturers use standard deviations to ensure their products meet certain specifications. If a product's measurement falls outside a certain number of standard deviations from the mean, it might indicate a problem in the manufacturing process. Similarly, in finance, standard deviation is used to measure the volatility of investments. A higher standard deviation indicates a higher level of risk.
The z-score for z 37 is calculated to be approximately 2.22, indicating that it is between 2 and 3 standard deviations from the mean. Thus, the correct statement is that z 37 is between 2 and 3 standard deviations of the mean.
;
