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# The Law of Large Numbers and the Central Limit Theorem Practice Questions

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Updated on Oct 5, 2011

To review these concepts, go to The Law of Large Numbers and the Central Limit Theorem Study Guide.

## The Law of Large Numbers and the Central Limit Theorem Practice Questions

### Practice

The receptionist believes that the doctor he works for requires the patients to wait too long. He randomly selects 100 patients over a six-week period and records the time each waits before being taken back to see the doctor. The receptionist finds that the average wait time was 20 minutes, and the sample standard deviation was 15 minutes.

1. Assuming a normal distribution and using the information given, sketch the approximate distribution of the number of minutes a randomly selected patient spent waiting to see the doctor.
2. Based on the graph in the previous practice question, explain why the population distribution is very unlikely to be normal and why it is most likely skewed to the right.
3. The receptionist believes that 20 minutes is close to the mean time patients spent waiting to see the doctor. Do you agree? Provide a justification for your response.
4. Based on samples of size 100, what is the approximate sampling distribution of the mean number of minutes patients spent waiting to see the doctor?

#### Solutions

1.

2. For a normal distribution, 95% of the population is within two standard deviations of the mean. However, in this example, two standard deviations either direction from the mean, we have that 95% of the population is between – 10 and 50. It is impossible to wait a negative number of hours. Thus, the distribution is most likely skewed right.
3. By the Law of Large Numbers, the sample mean of 20 is likely to be "close" to the population mean. (A sample size of 100 is usually considered to be large.)
4. Normal with mean μ and standard deviation

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