a) To find the probability that a problem can be solved in less than 12 minutes, we first need to convert 12 minutes into a z-score. The z-score is calculated by subtracting the mean from the value and dividing by the standard deviation.

Z = (X - μ) / σ

Where:
X = value (12 minutes)
μ = mean (17 minutes)
σ = standard deviation (3 minutes)

So, Z = (12 - 17) / 3 = -1.67

We then look up this z-score in a standard normal distribution table or use a calculator that can calculate the cumulative probability. The cumulative probability associated with Z = -1.67 is approximately 0.0475. So, the probability that a problem can be solved in less than 12 minutes is 0.0475, or 4.75%.

b) To find the probability that a problem can be solved between 18 and 20 minutes, we first convert 18 and 20 minutes into z-scores.

Z1 = (18 - 17) / 3 = 0.33
Z2 = (20 - 17) / 3 = 1

We then find the cumulative probabilities associated with these z-scores. The cumulative probability for Z1 = 0.33 is approximately 0.6293 and for Z2 = 1 is approximately 0.8413.

The probability that a problem can be solved between 18 and 20 minutes is the difference between these two probabilities, which is 0.8413 - 0.6293 = 0.212, or 21.2%.

c) If 40% of customers are dissatisfied with the waiting time, this corresponds to the 60th percentile of the distribution (since 100% - 40% = 60%). We need to find the waiting time associated with the 60th percentile.

We look up the z-score associated with the 60th percentile in a standard normal distribution table, which is approximately 0.25. We then convert this z-score back into a waiting time.

X = Zσ + μ

X = 0.25 * 3 + 17 = 17.75

So, the minimum waiting time for a dissatisfied customer is approximately 17.75 minutes.

Question

a) To find the probability that a problem can be solved in less than 12 minutes, we first need to convert 12 minutes into a z-score. The z-score is calculated by subtracting the mean from the value and dividing by the standard deviation.

Z = (X - μ) / σ

Where:
X = value (12 minutes)
μ = mean (17 minutes)
σ = standard deviation (3 minutes)

So, Z = (12 - 17) / 3 = -1.67

We then look up this z-score in a standard normal distribution table or use a calculator that can calculate the cumulative probability. The cumulative probability associated with Z = -1.67 is approximately 0.0475. So, the probability that a problem can be solved in less than 12 minutes is 0.0475, or 4.75%.

b) To find the probability that a problem can be solved between 18 and 20 minutes, we first convert 18 and 20 minutes into z-scores.

Z1 = (18 - 17) / 3 = 0.33
Z2 = (20 - 17) / 3 = 1

We then find the cumulative probabilities associated with these z-scores. The cumulative probability for Z1 = 0.33 is approximately 0.6293 and for Z2 = 1 is approximately 0.8413.

The probability that a problem can be solved between 18 and 20 minutes is the difference between these two probabilities, which is 0.8413 - 0.6293 = 0.212, or 21.2%.

c) If 40% of customers are dissatisfied with the waiting time, this corresponds to the 60th percentile of the distribution (since 100% - 40% = 60%). We need to find the waiting time associated with the 60th percentile.

We look up the z-score associated with the 60th percentile in a standard normal distribution table, which is approximately 0.25. We then convert this z-score back into a waiting time.

X = Zσ + μ

X = 0.25 * 3 + 17 = 17.75

So, the minimum waiting time for a dissatisfied customer is approximately 17.75 minutes.

Knowee AI · Accepted Answer