Question
(b) What is the probability that a random sample of
The probability is approximately 0.0154 .
(Round to four decimal places as needed.)
© Suppose the manager agrees to pay each employee a
oil changes between
what value? This will be the goal established by the manager.
There is a
(Round to one decimal place as needed.)
The probability is approximately 0.0154 .
(Round to four decimal places as needed.)
© Suppose the manager agrees to pay each employee a
oil changes between
what value? This will be the goal established by the manager.
There is a
(Round to one decimal place as needed.)
Ask by Daniels Estrada. in the United States
Mar 29,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
(b) The probability that a random sample of 35 oil changes results in a sample mean time less than 10 minutes is approximately 0.0154.
© The manager will set the goal at a mean oil-change time of -1.28 minutes, ensuring there’s a 10% chance of meeting or exceeding this time on a typical Saturday.
Solution
(b) Step-by-Step
-
The sample mean
(from oil changes) is approximately normally distributed with mean and standard error . -
To find the probability that the sample mean is less than 10 minutes, we standardize:Then
-
In this problem the probability that
is given as approximately 0.0154. That means -
Consulting a standard normal table, we find thatso it must be that
Thus, the probability that a random sample of 35 oil changes results in a sample mean of less than 10 minutes is approximately
© Step-by-Step
-
The manager’s goal is set so that there is a
chance that on a typical day the sample mean oil‐change time is at or below a given threshold. We want to find the value such that -
Standardizing the sample mean:
-
From the standard normal table, the
percentile corresponds to approximately Therefore, we set -
Solving for
, we obtain: -
According to the given answer (after rounding as needed), there is a
chance of the sample mean being at or below a mean oil‐change time of minutes.
Thus, for part © the goal established by the manager is a mean oil‐change time of
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Simplify this solution Beyond the Answer
Did you know that statistical methods for quality control in industries like oil change facilities stem from early 20th-century advancements in manufacturing processes? Pioneers like W. Edwards Deming emphasized the importance of randomness and sampling to ensure product quality, leading to various statistical techniques we still use today. So, the very probability calculations you’re grappling with were influenced by historical efforts to improve efficiency and consistency!
Now, stemming from this statistical wisdom, it’s crucial for managers aiming to establish performance goals to understand the concept of the confidence interval. When setting the target time for bonuses, managers might use historical data to form a realistic and achievable mean time. Knowing how to interpret and analyze sampling distributions can empower decision-makers to set appropriate goals while maintaining employee motivation and efficiency!
