Question

The amount of time spent by a statistical consultant with a client at their first meeting is a random variable having a normal distribution with a mean value of \(60 \mathrm{~min}\) and a standard deviation of \(10 \mathrm{~min}\). a. What is the probability that more than \(45 \mathrm{~min}\) is spent at the first meeting? b. What amount of time is exceeded by only \(10 \%\) of all clients at a first meeting? c. If the consultant assesses a fixed charge of \(\$ 10\) (for overhead) and then charges \(\$ 50\) per hour, what is the mean revenue from a client's first meeting?

Short answer

(a) The probability that more than 45 min is spent at the first meeting is 0.9332. (b) The amount of time at the first meeting exceeded by only 10% of all clients is approximately 72.85 min. (c) The mean revenue from a client's first meeting is approximately \$59.98.

Step-by-step solution

Calculate Probability for Part (a)

To find the probability that more than 45 minutes is spent at the first meeting, the Z-score needs to be calculated first, which is done by subtracting the mean from the value of interest, then dividing by the standard deviation. Hence, the Z-score Z1 is \( \frac{45-60}{10}=-1.5 \). This Z-score corresponds to a cumulative probability of 0.0668 according to the z-table. We need to find the probability of spending more than 45 minutes, which is obtained by subtracting this cumulative probability from 1. Hence, the answer is \(1 - 0.0668 = 0.9332 \).

Calculate Time for Part (b)

We need to find the time that is exceeded by only 10% of all clients at a first meeting. This means that we need to find the Z-score that corresponds to a cumulative probability of 0.90 (since 100% - 10% = 90%). According to the Z-table, this is 1.285. We then use this to find the corresponding time, denoted as T, by multiplying the Z-score by the standard deviation and adding the mean. Hence, \(T = 1.285 \cdot 10 + 60 = 72.85\) minutes.

Calculate Revenue for Part (c)

The consultant charges $50 per hour, which is equivalent to $50 / 60 = $0.833 per minute. Using the mean meeting time of 60 minutes, the revenue made from the hourly charge would be \(60 \times 0.833 = \$49.98\). Adding the fixed charge of $10, the total mean revenue from a client's first meeting is \(\$49.98 + \$10 = \$59.98\).