Download Distribution of the Sample Mean

Distribution of the Sample Mean
We selected Q8.1.28 (p.378) as an example of using StatCrunch to calculate probability of x .
Q8.1.28
Burger King’s Drive-Through Suppose that cars arrive at Burger King’s drive-through at the rate of 20 cars
every hour between 12:00 noon and 1:00 P.M. A random sample of 40 one-hour time periods between 12:00
noon and 1:00 P.M. is selected and has 22.1 as the mean number of cars arriving.
(a) Why is the sampling distribution of x approximately normal?
(b) What is the mean and standard deviation of the sampling distribution of x assuming that   20 and
  20 ?
(c) What is the probability that a simple random sample of 40 one-hour time periods results in a mean of at
least 22.1 cars? Is this result unusual? What might we conclude?
(a) Why is the sampling distribution of x approximately normal?
Since the sample size is large ( n  40  30 ), the Central Limit Theorem stated that the sampling
distribution of the mean, x , is approximately normal.
(b) What is the mean and standard deviation of the sampling distribution of x assuming that   20 and
  20 ?
The mean of sampling distribution x is x and x    20 .
The standard deviation of the sampling distribution of x is  x and  x 

n

40
20
 0.707106781  0.707
(c) What is the probability that a simple random sample of 40 one-hour time periods results in a mean of at
least 22.1 cars? Is this result unusual? What might we conclude?
---> Find P( x  22.1) .
x is normally distributed with x =20 and  x 0.707.
Step 1: Log onto StatCrunch and get a blank data sheet.
Step 2: Click Stat → Calculators → Normal.
Step 3: 1) 1) When the normal distribution dialogue box pops up. Click the Standard tab.
2) For this x variable, input Mean=20 for x and input Std. Dev. =0.707 for  x .
3) Use  to select  → Input 22.1
4) Click Compute.
The shaded area (a tiny red) represents P( x  22.1) . Since P( x  22.1)  0.00148765  0.0015  0.002
which is extremely small. This result is extremely unusual.
Note: 0.002 = 2 out of 1000
Conclusion:
If we take 1000 simple random samples of 40 one-hour time periods between 12:00 noon
and 1:00 p.m., about 2 of the samples will result in a mean average of at least 22.1 cars arriving
at the drive-through. This result is unusual because the probability is 0.2%. Therefore, the business
has not increased much for the one-hour time period between 12:00 noon and 1:00 p.m..