Download 5.3 The Sampling Distribution of the Sample Mean ̅ and the

5.3 The Sampling Distribution of the Sample Mean ̅ and the
Central Limit Theorem
Example (Ex 5.6, p. 277). Suppose a population has the uniform
probability distribution given in the figure. The mean and standard
deviation of this probability distribution are μ=175 and σ=14.43. Now
suppose a sample of 11 measurements is selected from this population.
Describe the sampling distribution of the sample mean ̅ based on the
1,000 simulations.
Simulation:
1. Select 11 independent observations
2. Compute
from the uniform distribution over the interval [150,200]
̅
3. Repeat simulation 1000 times so we get 1000 (different) sample means
4. Compute the sample mean and the sample standard deviation of these sample means
5. Make a histogram of these sample means
SOLUTION (Excel)
( ̅)
The mean of the sample mean ̅ ,
The standard deviation of the sample mean
̅
̅:
≈ ……
̅
………
Mean and the standard deviation of the sample mean ̅
Example (Ex 5.6, p. 277) cont.
Compute the mean and standard deviation of the sampling distribution in the previous example are
̅
̅
√
√
Sampling distribution of the sample mean ̅
1. Theorem 5.1 [normal population, any sample size (n)]. Consider a random sample of
n observations selected from a population with a normal distribution with mean µ and
standard deviation σ (no assumptions about sample size n). Then, for any n the
sampling distribution of ̅ is normal with mean
and standard deviation
̅
̅
√
.
2. Central Limit Theorem 5.2 [arbitrary population, large sample size (n)]. Consider a
random sample of n observations selected from any population (no assumptions
about population distribution) with mean µand standard deviation σ. Then, when n is
sufficiently large, the sampling distribution of ̅ is approximately normal with mean
̅
and standard deviation
̅
√
.
NOTE: n ≥ 30 is regarded sufficiently large. If n is small and a population distribution is far from
normal, then nothing can be said about the distribution of ̅
Example (Ex. 5.7, p. 279). Suppose we have selected a random sample of n = 36 observations from a population
with mean equal to 80 and standard deviation equal to 6.
a. What is an approximate distribution of the sample mean
ANS. Approximately normal with mean = 80 and standard deviation =
.
b. Compute the probability that
ANS. P( ̅
)
̅ will be larger than 82
(82,10^99,80,1) = .0228
√
Exercise 1. Given population distribution, population mean µ, population standard deviation σ, and
the sample size n, what can you say about the distribution of the sample mean ̅
a. µ = 110, σ = 15, n = 45 →
……,
̅
……, distribution of ̅ ………………………
̅
b. µ = 75, σ = 7, n = 8, population normal →
̅
……,
c. µ = 3.7, σ = 0.9, n = 11, population heavily skewed →
d. µ = 110, σ = 15, n = 250 →
̅
……,
̅
̅
̅
……, distribution of ̅ …………………
……,
……, distribution of
̅
……, distribution of ̅ …………
̅ ………………………
Exercise 3 [5.24, p. 284] According to a National Business navel Association (NBTA) 2010 survey,
the average salary of a travel management professional is $96,850. Assume that the standard deviation of such
salaries is $30,000. Consider a random sample of 50 travel management professionals and let ̅ represent the mean
salary for the sample.
a. What is
̅ ?
b. What is ̅
c. Describe the shape of the sampling distribution of ̅
d. Find the z-score for the value ̅ = 89,500.
e. Find P ( ̅ > 89,500).
Exercise 4. Statistics from a weather center indicate that a certain city receives an average of 25 inches of snow each
year, with a standard deviation of 7 inches. Assume that amount of snow in a year is normally distributed. A student
lives in this city for 4 years. Let ̅ represent the mean amount of snow for those 4 years.
a. Describe the sampling distribution model of this sample mean.
b. What is the probability that the mean amount of snow for those 4 years exceeds 30 inches.