Business Context
FinServe tracks customer support call durations to staff its service team. An operations analyst claims call times are approximately normal and wants to use normal-distribution properties for forecasting and service-level planning.
Problem Statement
Assume individual call durations follow a normal distribution with mean 8.4 minutes and standard deviation 1.6 minutes. Use the properties of the normal distribution to quantify typical ranges, tail probabilities, and the behavior of sample means.
Given Data
| Metric | Value |
|---|
| Population mean call duration, μ | 8.4 minutes |
| Population standard deviation, σ | 1.6 minutes |
| Threshold 1 | 10.0 minutes |
| Threshold 2 | 5.2 minutes |
| Sample size for daily average | 36 calls |
| Significance level for interval reference | 0.05 |
Requirements
- State the key properties of the normal distribution relevant to this problem.
- Compute the probability that a single call lasts more than 10.0 minutes.
- Compute the probability that a single call lasts between 5.2 and 11.6 minutes.
- Compute the probability that the average duration of 36 calls exceeds 9.0 minutes.
- Use the empirical rule to state the approximate ranges covering 68%, 95%, and 99.7% of individual call durations.
- Briefly explain why the sample mean is also normally distributed here.
Assumptions
- Call durations are independently distributed.
- The population distribution of call duration is normal.
- The stated mean and standard deviation are known and stable during the measurement period.
