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# Normal Distribution and Central Limit Theorem

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The mean amount purchased by a typical customer at Churchill's Grocery Store is \$23.50 with a standard deviation of \$5.00. Assume the distribution of amounts purchased follows the normal distribution. For a sample of 50 customers, answer the following questions.

a. What is the likelihood the sample mean is at least \$25.00?
b. What is the likelihood the sample mean is greater than \$22.50 but less than \$25.00?
c. Within what limits will 90 percent of the sample means occur?

https://brainmass.com/statistics/central-limit-theorem/normal-distribution-and-central-limit-theorem-231004

#### Solution Preview

The mean amount purchased by a typical customer at Churchill's Grocery Store is \$23.50 with a standard deviation of \$5.00. Assume the distribution of amounts purchased follows the normal distribution. For a sample of 50 customers, answer the following questions.

a. What is the likelihood the sample mean is at least \$25.00?

Mean=M = \$23.50
Standard deviation =s= \$5.00
sample size=n= 50
sx=standard error of mean=s/square root of n= 0.7071 = ( 5 /square root of 50)
xbar= \$25.00
z=(xbar-M )/sx= 2.1213 =(25-23.5)/0.7071
Cumulative Probability corresponding to z= ...

\$2.19

## Finite math/business analysis Multiple choice - normal distribution, probability distribution and Central Limit Theorem

If a particular batch of data is approximately normally distributed, we would find that approximately:
a. 2 of every 3 observations would fall between ±1 standard deviation around the mean.
b. 4 of every 5 observations would fall between ±1.28 standard deviations around the mean.
c. 19 of every 20 observations would fall between ±2 standard deviations around the mean.
d. All of the above are true.

14. The Tampa International Airport (TIA) has been criticized for the waiting times associated with departing flights. While the critics acknowledge that many flights have little or no waiting times, their complaints deal more specifically with the longer waits attributed to some flights. The critics are interested in showing, mathematically, exactly what the problems are. Which type of distribution would best model the waiting times of the departing flights at TIA?
a. Binomial distribution
b. Poisson distribution
c. Normal distribution
d. Exponential distribution

15. The Central Limit Theorem is important in statistics because.
a. for a large n, it says the population is approximately normal.
b. for any population, it says the sampling distribution of the sample mean is approximately normal, regardless of the sample
size.
c. for a large n, it says the sampling distribution of the sample mean is approximately normal, regardless of the shape of the
population.
d. for any sized sample, it says the sampling distribution of the sample mean is approximately normal.

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