Use the following information to answer the next three exercises: The length of time it takes to find a parking space at 9 a.m. follows a normal distribution with a mean of five minutes and a standard deviation of two minutes.
Thuy Dau, Ngoc Bui, Sam Su, and Lan Voung conducted a survey as to how long customers at Lucky claimed to wait in the checkout line until their turn. Let X = time in line. the table displays the ordered real data, in minutes.
| 0.50 | 4.25 | 5 | 6 | 7.25 |
| 1.75 | 4.25 | 5.25 | 6 | 7.25 |
| 2 | 4.25 | 5.25 | 6.25 | 7.25 |
| 2.25 | 4.25 | 5.5 | 6.25 | 7.75 |
| 2.25 | 4.5 | 5.5 | 6.5 | 8 |
| 2.5 | 4.75 | 5.5 | 6.5 | 8.25 |
| 2.75 | 4.75 | 5.75 | 6.5 | 9.5 |
| 3.25 | 4.75 | 5.75 | 6.75 | 9.5 |
| 3.75 | 5 | 6 | 6.75 | 9.75 |
| 3.75 | 5 | 6 | 6.75 | 10.75 |
- Calculate the sample mean and the sample standard deviation.
- Construct a histogram.
- Draw a smooth curve through the midpoints of the tops of the bars.
- In words, describe the shape of your histogram and smooth curve.
- Let the sample mean approximate μ and the sample standard deviation approximate σ. The distribution of X can then be approximated by X ~ _____ (_____, _____)
- Use the distribution in part e to calculate the probability that a person will wait fewer than 6.1 minutes.
- Determine the cumulative relative frequency for waiting less than 6.1 minutes.
- Why aren’t the answers to part f and part g exactly the same?
- Why are the answers to part f and part g as close as they are?
- If only 10 customers were surveyed rather than 50, do you think the answers to part f and part g would have been closer together or farther apart? Explain your conclusion.
OpenStax Statistics, 6.2 Using the Normal Distribution, exercise 51