The set of life spans of an appliance is normally distributed with a mean [tex]$\mu=48$[/tex] months and a standard deviation [tex]$\sigma=$[/tex] 8 months. What is the [tex]$z$[/tex]-score of an appliance that stopped working at 64 months?
A. -2
B. -1
C. 1
D. 2
Answer (2)
The problem provides the mean, standard deviation, and a specific value from a normal distribution.
The z -score formula is z = σ x − μ .
Substituting the given values, z = 8 64 − 48 .
Calculating the z -score gives 2 .
Explanation
Understand the problem and provided data We are given that the life spans of an appliance are normally distributed with a mean μ = 48 months and a standard deviation σ = 8 months. We want to find the z -score of an appliance that stopped working at x = 64 months.
Recall the formula for calculating the z-score The z -score is a measure of how many standard deviations away from the mean a particular data point is. The formula for calculating the z -score is: z = σ x − μ where:
x is the observed value,
μ is the mean of the distribution,
σ is the standard deviation of the distribution.
Substitute the given values into the formula Now, we substitute the given values into the formula: z = 8 64 − 48
Calculate the z-score Let's calculate the z -score: z = 8 16 = 2 So, the z -score of an appliance that stopped working at 64 months is 2.
State the final answer The z -score of an appliance that stopped working at 64 months is 2 . This means that the appliance's lifespan is 2 standard deviations above the mean lifespan.
Examples
Understanding z-scores is crucial in many real-world scenarios. For instance, in quality control, if a machine produces items with a certain specification (e.g., length), the z-score can help determine how far a particular item's measurement deviates from the average. If the z-score is too high or too low, it indicates a problem with the manufacturing process, allowing engineers to make necessary adjustments to maintain quality standards. Similarly, in finance, z-scores can be used to assess the risk associated with an investment by measuring how far its returns deviate from the average market return.
The z -score of an appliance that stopped working at 64 months is 2, meaning it is two standard deviations above the mean lifespan of 48 months. Therefore, the correct answer is 2. This indicates the appliance performed better than average in terms of lifespan.
;
