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(Lesson 24: Point Estimates of Parameters) 24.
ESTIMATING POPULATION PARAMETERS
How do we estimate population parameters such as a population mean, standard
deviation (SD), variance, or proportion?
LESSON 24: POINT ESTIMATES OF PARAMETERS
What is Our Best Guess?
PART A: INFERENTIAL STATISTICS
Here is the overview from Lesson 1:
A Statistical Experiment Probability
- Designing an Experiment |
- Collecting Data |
- Describing Data |
(Descriptive Statistics) |
- Interpreting Data using ← − − − − − −
(Inferential Statistics)
Size: N elements (or members)
Population [of interest] All adult Americans?
All registered voters in California?
Size: n elements (or members)
Sample For a poll? A scientific study?
Are we testing products for quality control?
We now enter the realm of inferential statistics, the science of interpreting data.
What can a sample tell us about the population from which it is drawn?
(Lesson 24: Point Estimates of Parameters) 24.
PART B: POINT ESTIMATES
If a single value is used to estimate a population parameter, we call that value a
point estimate for that parameter.
Let’s say we draw a sample from a population.
Mean SD VAR
Population (Size N ) μ σ σ
2
Sample (Size n )
x s s
2
We use sample statistics to estimate population parameters.
- We will use the sample mean , x , as a point estimate for the population
mean , μ.
- We will use the sample standard deviation , s , as a point estimate for the
population standard deviation , σ.
- We will use the sample variance , s
2
, as a point estimate for the
population variance , σ
2
Example 1 (Point Estimates)
A large lecture class takes a test. Five of the tests are randomly selected and
graded. The grader reports that x = 75 points , s = 15 points , and
s
2
= 225 square points.
- a) What is a point estimate for the population mean of test scores for the
entire class?
- b) What is a point estimate for the population standard deviation of test
scores for the entire class?
- c) What is a point estimate for the population variance of test scores for
the entire class?
(Lesson 24: Point Estimates of Parameters) 24.
§ Solution
- a) What is a point estimate for the population mean of test scores for the
entire class?
The point estimate is the sample mean , x = 75 points.
- b) What is a point estimate for the population standard deviation of test
scores for the entire class?
The point estimate is the sample standard deviation , s = 15 points.
- c) What is a point estimate for the population varianc e of test scores for
the entire class?
The point estimate is the sample variance , s
2
= 225 square points.
PART C: UNBIASED ESTIMATORS
A sample statistic is an unbiased estimator for a population parameter when the
expected value of the sample statistic is the value of the population parameter.
A key reason why we use the sample mean as a point estimate for the population
mean is that the sample mean is an unbiased estimator for the population mean:
E X
= μ.
- What does that mean? Consider all possible samples of size five from the
class … and all of their sample means (values of X ). The average of those
sample means would be the population mean , μ.
- Unbiased estimators do not have a tendency to overestimate or
underestimate the population parameter.
The sample variance is an unbiased estimator for the population variance.
The sample standard deviation is not an unbiased estimator for the population
standard deviation , but it is still good enough to use as a point estimate.
(Lesson 24: Point Estimates of Parameters) 24.
PART D: SAMPLE PROPORTIONS
p denotes a population proportion.
- It could represent a probability , such as the success probability p from a
Bin n , p
distribution.
p ˆ denotes a sample proportion. It is an unbiased estimator of p.
Proportion
Population (Size N ) p
Sample (Size n )
p
Calculating a Sample Proportion
A success is a property that we are interested in.
If n is a sample size , and if x is the number of successes in the sample,
then the sample proportion of successes is given by:
p =
x
n
If n is the number of trials in a binomial experiment , and if
x is the number of “successful” trials , then our point estimate for p ,
the success probability per trial in a Bin n , p
distribution, is given by:
p =
x
n
This is the sample proportion of successes among the n trials.
(Lesson 25: Confidence Interval Estimates of Parameters) 25.
LESSON 25: CONFIDENCE INTERVAL ESTIMATES
OF PARAMETERS
What if We’re Allowed a Range of Guesses?
PART A: SAMPLING ERROR
A sample often does not represent the population (or a probability) perfectly, and
this leads to sampling error.
A point estimate , typically a sample statistic such as the sample mean x , may or
may not equal the true value of the population parameter that is being estimated.
Example 1 (Sampling Error)
- a) The population mean for a class exam could be 75 points,
while the sample mean for five randomly selected exams from that class
could be 74 points. That is, μ = 75 points , while x = 74 points.
- b) A fair coin may come up heads 51 times in 100 flips. If p is the
probability that the coin comes up heads, then p = 0.5 for a fair coin, while
the sample proportion p ˆ = 0.51.
(Lesson 25: Confidence Interval Estimates of Parameters) 25.
PART B: INTERVAL ESTIMATES
To account for sampling error, we will provide a range of values, called an interval
estimate for a population parameter, such as a population mean.
Example 2 (Interval Estimate for a Population Mean)
A lecture class takes an exam. We want an interval estimate for μ , the
population mean of exam scores in the class. A random sample of five
exams is selected and graded. Our interval estimate for μ could be the
purple interval
70 points, 78 points ( ) below.
- The lower limit of the interval estimate is 70 points.
- The upper limit of the interval estimate is 78 points.
- The sample mean x is at the center (midpoint) of the interval.
It is the average of the lower limit and the upper limit:
x =
lower limit ( )
= 74 points
- The interval estimate is symmetric about the point estimate
(here, x = 74 points ). This will not always be the case; we will use
asymmetric intervals when estimating standard deviations and variances.
(Lesson 25: Confidence Interval Estimates of Parameters) 25.
PART C: MARGIN OF ERROR
If an interval estimate is symmetric about the point estimate, then we may write
the interval in terms of the point estimate and the margin of error.
Margin of Error of a Symmetric Interval Estimate
- The margin of error (denoted by E ) is the distance between the
point estimate and either limit of the interval.
- It is half the width of the entire interval.
- It is the maximum likely distance between the point estimate and
the true value of the population parameter being estimated.
Example 3 (Margin of Error; Revisiting Example 2)
In Example 2, we estimated the population mean of exam scores in a class.
Our point estimate was the sample mean x = 74 points. Our interval
estimate
70 points, 78 points ( ) was symmetric about the point estimate.
There are three ways to calculate the margin of error E :
− x = 78 − 74 = 4 points
= 74 − 70 = 4 points
- It is half the width of the entire interval:
E =
upper limit ( )
− lower limit ( )
= 4 points
The interval estimate (70 poi nts, 78 points) for the population mean μ
can be written in terms of the sample mean x and the margin of error E :
μ = x ± E
μ = 74 ± 4 in points ( )
We could informally say that we believe that the population mean is about
74 points, give or take 4 points.
(Lesson 25: Confidence Interval Estimates of Parameters) 25.
PART D: CONFIDENCE INTERVAL ESTIMATES
We believe that an interval estimate for a population parameter (such as the
population mean μ ) is likely to contain the value of that parameter.
If we attach a confidence level to the interval estimate, then we have a
confidence interval (“CI”) estimate for the parameter.
A confidence level is a probability that is often expressed as a percent.
- It is the probability that the confidence interval contains the true value of
the parameter.
- The most commonly used confidence level is 95%. This is the typically
assumed confidence level in published studies and news reports.
Example 4 (Confidence Levels and Confidence Intervals)
Let μ be the population mean I.Q. score of American adults.
A psychology professor wants to estimate μ. The professor analyzes
a sample of American adult I.Q.s. The sample mean x = 101 points
and a 95% confidence interval (CI) for μ is 98 points, 104 points ( )
Interpret this confidence interval (CI).
§ Solution
We are 95% confident that this interval contains the population mean I.Q.
score of American adults.
- Some books (such as Triola’s) object to saying that “there is a 95%
chance” that μ “falls in the confidence interval.” See the next page …
(Lesson 26: z , t , and χ
2
Distributions) 26.
LESSON 26 : z , t , and χ
2
DISTRIBUTIONS
What Distributions Will Help Us Construct Confidence Intervals?
PART A: REVIEW OF STANDARD NORMAL ( z ) DISTRIBUTIONS
The standard normal ( z ) distribution is the normal distribution with mean 0 and
standard deviation 1 :
Z ~ N μ = 0 , σ = 1 ( )
Properties of Standard Normal ( z ) Distributions
- They are continuous.
- A normal random variable can take on any real number as a value.
- The density curve is bell-shaped.
- The total area under the density curve (and above the horizontal axis)
is 1. This is true for all probability density curves.
- The mean is 0.
- The density curve is symmetric about the mean.
- The standard deviation is 1.
- The standard deviation is the distance between the “mean-line” and
either inflection point (“IP”); “IP”s are points where the density curve
changes from concave up (curving upward) to concave down (curving
downward), or vice-versa.
- The tails of the curve fall off rapidly (in terms of standard deviations).
General Normal Distribution Standard Normal Distribution
(Lesson 26: z , t , and χ
2
Distributions) 26.
PART B: t DISTRIBUTIONS
t distributions, also called Student’s t distributions, are similar to the standard
normal ( z ) distribution.
Similarities Between t and z Distributions
- They are continuous.
- A random variable can take on any real number as a value.
- The density curve is bell-shaped.
- The total area under the density curve is 1.
- The mean is 0.
- The density curve is symmetric about the mean.
- The tails of the curve fall off rapidly (in terms of standard deviations).
Differences Between t and z Distributions
- There are infinitely many different t distributions.
- There is only one z distribution.
- For a z distribution, the standard deviation σ = 1.
- For a t distribution, the standard deviation σ > 1.
- The tails of a “ t density curve” do not fall off as rapidly as for the
“ z curve.”
- The two inflection points (IPs) for a t distribution aren’t as
meaningful as for a z distribution. (See Footnote 1.)
(Lesson 26: z , t , and χ
2
Distributions) 26.
Degrees of Freedom (df) for a t Distribution
- There are different t distributions corresponding to different numbers
of degrees of freedom (df).
- The number of df can be any positive integer: 1, 2, 3, etc.
- For most basic applications, if n is the sample size, then we want to use
the t distribution on n − 1 ( )
df.
- As the number of df increases , a t distribution looks more like the z
distribution. The tails get thinner. The standard deviation σ decreases
and approaches 1, which is the standard deviation of the z distribution.
- If the number of df is large , a t distribution is often approximated by
the z distribution.
In the figure below:
- The orange , dashed curve represents the t distribution on 1 df.
- The blue , dotted curve represents the t distribution on 5 df.
- The black , solid curve represents the z distribution.
(Lesson 26: z , t , and χ
2
Distributions) 26.
PART C : χ
2
DISTRIBUTIONS
χ
2
distributions , written out as “chi-square distributions”, have both similarities
and key differences compared to the z and t distributions.
Similarities that χ
2
Distributions Have with z and t Distributions
- They are continuous.
- The total area under the density curve is 1.
- The number of degrees of freedom (df) for a χ
2
distribution has
some similarities to the number of df for a t distribution.
Degrees of Freedom (df) for a χ
2
Distribution
2
distributions corresponding to different
numbers of degrees of freedom (df).
- The number of df can be any positive integer: 1, 2, 3, etc.
- For most basic applications, if n is the sample size, then we want to use
the χ
2
distribution on n − 1 ( )
df.
- As the number of df increases , a χ
2
distribution looks more like a
normal distribution, though not the standard normal z distribution
(which the t distributions could resemble).
- If the number of df is large , a χ
2
distribution is often approximated
by a normal distribution , though not the z distribution.
