Example 7.9 from Statistics, 7.3 Using the Central Limit Theorem

(part 1)

Using the clt to find probability

Find the probability that the mean excess time used by the 80 customers in the sample is longer than 20 minutes. This is asking us to find P( $\bar{x}$ > 20). Draw the graph.
Suppose that one customer who exceeds the time limit for his cell phone contract is randomly selected. Find the probability that this individual customer's excess time is longer than 20 minutes. This is asking us to find P(x > 20).
Explain why the probabilities in parts (a) and (b) are different.

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

1
Solution
1. Find: P( $\bar{x}$ > 20)
  
  P( $\bar{x}$ > 20) = 0.79199 using normalcdf $(20,1E99,22, \frac{22}{\sqrt{80}})$
  The probability is 0.7919 that the mean excess time used is more than 20 minutes, for a sample of 80 customers who exceed their contracted time allowance.
  
  Figure 7.8
  
  Reminder
  
  1E99 = 10⁹⁹ and –1E99 = –10⁹⁹. Press the EE key for E. Or just use 10⁹⁹ instead of 1E99.
2. Find P(x > 20). Remember to use the exponential distribution for an individual. $X ~ E x p (\frac{1}{22})$ .
  $P (x > 20) = e^{(- (\frac{1}{22}) (20))} or e^{(- 0 .04545(20))} = 0 .4029$
3. 1. P(x > 20) = 0.4029, but P( $\bar{x}$ > 20) = 0.7919
  2. The probabilities are not equal because we use different distributions to calculate the probability for individuals and for means.
  3. When asked to find the probability of an individual value, use the stated distribution of its random variable; do not use the clt. Use the clt with the normal distribution when you are being asked to find the probability for a mean.

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