Point Estimates and Sampling Variability
Elementary Statistics
MTH-161D | Spring 2025 | University of Portland
March 12, 2025
Objectives
- Introduce the Central Limit Theorem (CLT)
- Know how to determine a point estimate
- Develop an understanding of the sampling
distribution
- Activity: Understanding the CLT
These slides are derived from Diez et al.
(2012).
Previously… (1/3)
The guiding principle of statistics is statistical thinking.
Previously… (2/3)
Types of Inference
| Goal |
Estimate an unknown population value |
Assess claims about a population value |
| Methods |
Point Estimation: A single value estimate (e.g., sample
mean)
Interval Estimation: A range of plausible values (e.g.,
confidence interval) |
State a null and an alternative hypothesis
Compute
a test statistic and compare it to a threshold (p-value or critical
value) |
| Key Concept |
Focuses on precision in estimation (confidence
intervals) |
Focuses on decision-making based on evidence (reject or
fail to reject the null hypothesis) |
Previously… (3/3)
The normal r.v. \(X \sim
\text{N}(\mu,\sigma^2)\) has infinite possible outcomes (or
infinite sized sample space) where \(\mu\) is the mean and \(\sigma^2\) is the variance (\(\sigma\) is the standard deviation) with
PDF given the continuous curve below.
The Central Limit Theorem (CLT)
\(\star\) Key Idea:
CLT says that the sample mean (or sum) of many independent and
identically distributed random variables approaches a normal
distribution, regardless of the original distribution.
Parameter Estimation
- We are often interested in population
parameters.
- Since complete populations are difficult (or impossible) to collect
data on, we use sample statistics as point
estimates for the unknown population parameters of
interest.
- Sample statistics vary from sample to sample.
- Quantifying how sample statistics vary provides a way to estimate
the margin of error associated with our point
estimate.
Example 1
If we randomly sample 1,000 adults from each U.S. state, would the
sample means of their heights be:
- very different
- the same
- not the same, but only somewhat different
\(\star\) The answer is not the
same, but only somewhat different because of sampling
variability.
Example 2
Suppose the proportion of American adults who support the expansion
of solar energy is \(p = 0.88\).
- Is the provided \(p\) a population
parameter or a sample statistic?
- Is a randomly selected American adult more or less likely to support
the expansion of solar energy?
\(\star\) \(p=0.88\) is a population parameter because
it is talking about all american adults. The proportion is considered a
high proportion of support. Thus, a randomly selected american adult is
more likely to support solar energy expansion.
Example 2: Unknown Population Parameter
Suppose that you don’t have access to the population of all American
adults, which is a quite likely scenario. In order to estimate the
proportion of American adults who support solar power expansion, you
might sample from the population and use your sample proportion as the
best guess for the unknown population proportion.
- Sample, with replacement, 1000 American adults from the population,
and record whether they support solar power or not expansion.
- Find the sample proportion.
- Plot the distribution of the sample proportions.
\(\star\) Key Idea:
After many repeated sampling of the same process as described, the
resulting distribution of proportions will be normal.
Example 2: Point Estimate
\(\dagger\) Based on this
distribution, what do you think is the true population proportion?
Sampling Distributions
Sampling distributions are never observed
- In real-world applications, we never actually observe the
sampling distribution, yet it is useful to always think
of a point estimate as coming from such a
hypothetical distribution.
\(\star\) Key Idea:
Understanding the sampling distribution will help us characterize and
make sense of the point estimates that we do observe.
The Normal Distribution (Revisited)
The normal r.v. \(X \sim
\text{N}(\mu,\sigma^2)\) has infinite possible outcomes (or
infinite sized sample space) where \(\mu\) is the mean and \(\sigma^2\) is the variance (\(\sigma\) is the standard deviation) with
PDF given the continuous curve below.
The 68-95-99.7 Rule (1/3)
1st standard deviation from the mean
\[P(\mu - \sigma \le X \le \mu + \sigma)
\approx 0.68\]
The 68-95-99.7 Rule (2/3)
2nd standard deviation from the mean
\[P(\mu - 2\sigma \le X \le \mu + 2\sigma)
\approx 0.95\]
The 68-95-99.7 Rule (3/3)
3rd standard deviation from the mean
\[P(\mu - 3\sigma \le X \le \mu + 3\sigma)
\approx 0.997\]
Total Area Under the Curve
The Normal PDF satisfies the probability axioms
\[P(\mu - \infty \le X \le \mu + \infty)
\approx 1\]
\(\star\) Key Idea:
Because of the axiom that the sum of the probabilities for all outcomes
in the sample space is equal to 1, the total area under the Normal PDF
is always 1.
Standard Normal Distribution (1/2)
The standard normal distribution is when \(\mu=0\) and \(s=1\) or \(Z \sim
\text{N}(0,1)\).
The transformation formula (the z-score)
Standardized scores that measure how many standard deviations a value
is from the mean. \[Z = \frac{X -
\mu}{\sigma}\]
Standard Normal Distribution (2/2)
The standard normal distribution, \(Z \sim \text{N}(0,1)\).
\(\star\) Key Idea:
The standard normal distribution is that it is a normal distribution
with a mean of 0 and a standard deviation of 1. It serves as a reference
distribution, allowing any normally distributed variable to be
standardized.
CLT Conditions
- Independence – Sample values must be
independent
- Identical Distribution – Variables should be from
the same distribution
- Finite Variance – The population must have a finite
variance
- Large Sample Size – A larger sample size improves
approximation
Extending the Framework for other Descriptive Statistics
- The strategy of using a sample statistic to estimate a parameter is
quite common, and it’s a strategy that we can apply to other statistics
besides a proportion.
Example:
Take a random sample of students at a college and ask them how many
extracurricular activities they are involved in to estimate the average
number (or median number) of extra curricular activities all students in
this college are interested in.
\(\star\) Key Idea:
The principles and general ideas of CLT apply to other parameters as
well, even if the details change a little.
Activity: Understanding the CLT
- Make sure you have a copy of the W 3/12 Worksheet. This
will be handed out physically and it is also digitally available on
Moodle.
- Work on your worksheet by yourself for 10 minutes. Please read the
instructions carefully. Ask questions if anything need
clarifications.
- Get together with another student.
- Discuss your results.
- Submit your worksheet on Moodle as a
.pdf file.
