Questions: Your company has a contract to perform preventive maintenance on thousands of air-conditioning units in a large city. Based on service records from the past year, the time (in hours) that a technician requires to complete the work follows a strongly right-skewed distribution with μ=1 hour and σ=1.5 hours. As a promotion, your company will provide service to a random sample of 70 air-conditioning units free of charge. You plan to budget an average of 1.1 hours per unit for a technician to complete the work. Will this be enough time? Calculate the probability that the average maintenance time x̄ for 70 randomly selected units exceeds 1.1 hours.

Your company has a contract to perform preventive maintenance on thousands of air-conditioning units in a large city. Based on service records from the past year, the time (in hours) that a technician requires to complete the work follows a strongly right-skewed distribution with μ=1 hour and σ=1.5 hours. As a promotion, your company will provide service to a random sample of 70 air-conditioning units free of charge. You plan to budget an average of 1.1 hours per unit for a technician to complete the work. Will this be enough time?
Calculate the probability that the average maintenance time x̄ for 70 randomly selected units exceeds 1.1 hours.

Transcript text: Stacked Your company has a contract to perform preventive maintenance on thousands of air-conditioning units in a large city. Based on service records from the past year, the time (in hours) that a technician requires to complete the work follows a strongly right-skewed distribution with $\mu=1$ hour and $\sigma=1.5$ hours. As a promotion, your company will provide service to a random sample of 70 air-conditioning units free of charge. You plan to budget an average of 1.1 hours per unit for a technician to complete the work. Will this be enough time? ${ }_{20}$ Calculate the probability that the average maintenance time $\bar{x}$ for 70 randomly selected units exceeds 1.1 hours.

Solution

Solution Steps

Step 1: Define the Problem

We need to calculate the probability that the average maintenance time $ \bar{x} $ for 70 randomly selected air-conditioning units exceeds 1.1 hours. The population mean $ \mu $ is 1 hour, and the population standard deviation $ \sigma $ is 1.5 hours.

Step 2: Calculate Z-scores

To find the probability, we first calculate the Z-scores for the sample mean threshold of 1.1 hours. The Z-score is calculated using the formula:

\[ Z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}} \]

Substituting the values:

\[ Z_{start} = \frac{1.1 - 1}{1.5 / \sqrt{70}} \approx 0.5578 \]

The Z-score for the upper limit (infinity) is:

\[ Z_{end} = \infty \]

Step 3: Calculate the Probability

The probability that the sample mean exceeds 1.1 hours is given by:

\[ P(\bar{x} > 1.1) = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(\infty) - \Phi(0.5578) \]

Using the cumulative distribution function $ \Phi $:

\[ P(\bar{x} > 1.1) = 1 - 0.2885 = 0.2885 \]

Step 4: Interpret the Results

The calculated probability $ P $ that the average maintenance time exceeds 1.1 hours is approximately 0.2885. This indicates that there is a 28.85% chance that the average maintenance time will exceed the budgeted time of 1.1 hours.