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CHAPTER 7
z Introduction
• Parameters are numerical descriptive measures for populations.
– For the normal distribution, the location and shape are described by µ and σ.
– For a binomial distribution consisting of n trials, the location and shape are
determined by p.
• Often the values of parameters that specify the exact form of a distribution are unknown.
• You must rely on the sample to learn about these parameters.
Simple Random Sampling
• The sampling plan or experimental design determines the amount of information
you can extract, and often allows you to measure the reliability of your inference.
• Simple random sampling is a method of sampling that allows each possible sample
of size n an equal probability of being selected. In general, if a sample of n elements
is selected from a population of N elements using a sampling plan in which each of
the possible samples has the same chance of selection, then the sampling is said to be
random and the resulting sample is a simple random sampling.
The numerical descriptive measures that calculate from the sample are called statistics.
Sampling Distributions
The sampling distribution of a statistic is the probability distribution for the possible
values of the statistic that results when random samples of size n are repeatedly drawn from
the population.
Sampling distributions for statistics can be
• Derived mathematically using the laws of probability
• Approximated with simulation techniques
• Used statistical theorems to derive exact or approximate sampling distribution
The Central Limit Theorem is one such theorem.
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Central Limit Theorem
If random samples of n observations are drawn from a nonnormal population with finite
µ and standard deviation σ, then when n is large, the sampling distribution of the sample
√
mean x̄ is approximately normally distributed, with mean µ and standard deviation σ/ n.
The approximation becomes more accurate as n becomes large.
Why is this important?
• The Central Limit Theorem also implies that the sum of n measurements is ap√
proximately normal with mean nµ and standard deviation σ n.
• Many statistics that are used for statistical inference are sum or average of sample
measurements.
• When n is large, these statistics will have approximately normal distributions.
• This will allow us to describe their behavior and evaluate the reliability of our inferences.
How large is large?
• If the sample is normal, then the sampling distribution of x̄ will also be normal, no
matter what the sample size.
• When the sample population is approximately symmetric, the distribution becomes
approximately normal for relatively small values of n.
• When the sample population is skewed,the sample size must be at least 30 before
the sampling distribution of x̄ becomes approximately normal.
The Sampling Distribution of the Sample Mean x̄
• If a random sample of n measurements is selected from a population with mean µ and
standard deviation σ, the sampling distribution of sample mean x̄ will have mean µ
√
and standard deviation σ/ n.
• If the population has a normal distribution, the sampling distribution of x̄ will be
exactly normally distributed, regardless of the sample size, n.
• If the population distribution is nonnormal, the sampling distribution of x̄ will be
approximately normally distributed for large samples (by CLT).
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Standard Error
the standard deviation of a statistic used as an estimator of a population parameter is also
called the standard error of the estimator, SE because it refers to the precision of the
√
estimator. Therefore, the standard deviation of x̄- given by σ/ n- is referred to as the
standard error of the mean
Finding Probabilities for the Sample Mean
If the sampling distribution of x̄ is normal or approximately normal, standardize the interval
of interest in terms of
x̄ − µ
z= √
σ/ n
Find the appropriate area using table.
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Examples:
1. The soda filling machine is supposed to fill cans of soda with 12 fluid ounces. Suppose
that the fills are actually normally distributed with a mean of 12.1 oz and a standard
deviation of 2 oz. What is the probability that the average fill for a 6-pack of soda is
less than 12 oz?
2. Suppose a random sample of n = 25 observations is selected from a population that is
normally distributed, with mean equal to 106 and standard deviation equal to 12.
a. Give the mean and the standard deviation of the sampling distribution of the sample
mean x̄.
b. Find the probability that x̄ exceeds 110.
c. Find the probability that the sample mean deviates from the population mean
µ = 106 by no more than 4.
3. Suppose than college faculty with the rank of professor at 2-years institutions earn
an average of $65,608 per year with a standard deviation of $4000. In an attempt to
verify this salary level, a random sample of 60 professors was selected from a personnel
database for all 2-year institutions in the United States.
a. Describe the sampling distribution of the sample mean x̄.
b. Within what limits would you expect the sample average to lie, with probability
0.95?
c. Calculate the probability that the sample mean x̄ is greater than $67,000.
d. If your random sample actually produced a sample mean of $67,000, would you
consider this unusual? What conclusion might you draw?
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The Sampling Distribution of the Sample Proportion
• The Central Limit Theorem can be used to conclude that the binomial random variable x is approximately normal when n is large, with mean np and standard deviation
npq.
• The sample proportion, p̂ =
divided by n.
x
n
is simply a rescaling of the binomial random variable x,
• From the Central Limit Theorem, the sampling distribution of p̂ will also be approximately normal, with a rescaled mean and standard deviation
The Sampling Distribution of the Sample Proportion
• A random sample of size n is selected from a binomial population with parameter p.
• The sampling distribution of sample proportion, p̂ =
p
.
deviation SE(p̂) = pq
n
x
n
will have mean p and standard
• If n is large, and p is not too close to zero or one, the sampling distribution of p̂ will
be approximately normal
Finding Probabilities for the Sample Proportion
If the sampling distribution of p̂ is normal or approximately normal, standardize the interval
of interest in terms of
p̂ − p
z = p pq
n
Find the appropriate area using table
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Examples:
1. The soda bottler in the previous example claims that only 5% of the soda cans are
underfilled. A quality control technician randomly samples 200 cans of soda. What is
the probability that more than 10% of the cans are underfilled?
2. The percentage of students who used the internet as their major resource for a school
project in the past year was 66%. Sup pose that you take a sample of n = 100 students,
and record the number of students who used the Internet as their major resource for
their school project during the past year. Let p̂ be the proportion of students surveyed
who used the Internet as a major resource in the past year
a. What is the exact distribution of p̂? How can you approximate the distribution of
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p̂?
b. What is the probability that the sample proportion p̂ exceeds 68%?
c. What is the probability that the sample proportion lies between 64% and 68%?
d. Would a sample proportion of 70% contract the reported value of 66%?
———————————————————————————————Suggested Exercises: 7.19, 7.21, 7.27, 7.29, 7.31, 7.33, 7.37, 7.43, 7.45, 7.47, 7.63, 7.65,
7.73, 7.75, 7.77
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