Parameter Estimation
Applied Statistics
MTH-361A | Spring 2026 | University of Portland
Objectives
- Introduce confidence intervals
- Know how to compute confidence intervals
- Develop an understanding of interpreting confidence
intervals
Confidence Intervals
A plausible range of values for the population parameter is called a
confidence interval.
Fishing analogy:
A confidence interval is like using a fishing net, rather than a
spear, to catch fish in a murky lake.
- Parameter (the fish): The true population parameter (e.g.,
mean fish length). This is assumed to be fixed but unknown.
- Sample statistic (the catch): The observation (e.g., the
data collected from the lake) and the point estimate (e.g., sample mean
fish length).
The interval:
We need to set the size of the fishing net first before casting it
into the lake.
- Confidence level (the net size): A higher confidence level
(e.g., 99%) means a wider, larger net, increasing the chance of catching
the fish.
- Confidence interval (the net): The range of values
calculated from the sample.
\(\star\) If we report a point
estimate, we probably won’t hit the exact population parameter. If we
report a range of plausible values we have a good shot at capturing the
parameter.
Spam Emails
Suppose we want to estimate the number of spam emails of an
account.
Data summary:
- The data shows that there are \(16\) spam emails out of \(200\) emails.
- The population of interest is all emails of one account.
- The data we have is a random sample from the population.
Point estimate:
- The sample statistic is \[\overline{x} = 16,\] which is our
point estimate.
- \(\overline{x}\) is the number of
spam emails, an estimate to an unknown population
parameter.
\(\star\) For the purpose of this
example, the sample mean notation \(\overline{x}\) can be thought of as the
mean of a binomial r.v..
Sampling Distribution
The point estimate for the number of spam emails is \(\overline{x} = 16\) and the sample size is
\(n = 200\).
Assumptions:
- The emails are independently sampled from the same account
- The probability of getting a randomly selected spam email is
fixed.
- We assume that CLT conditions are satisfied.
- The sampling distribution of the number of spam emails exhibits a
binomial distribution.
Sampling distribution of \(\overline{x}\):
\(\star\) The domain of this
distribution is \(0 \le x \le 200\)
(the graph is truncated) because there are \(n
= 200\) samples and the \(x\)
value is the number of spam emails out of \(n\), where it could be \(0\), \(200\), or in between.
Confidence Interval of the Number of Spam Emails
The confidence level is a probability that we set. This is how wide
our interval is going to be.
Confidence Level:
Suppose we want a “\(90\)%
confidence interval”. So, the confidence level is \(0.90\).
This is the interval probability of the sampling
distribution.
The goal is to find \(a\) and
\(b\), so that \[P(a \le \overline{X} \le b) \approx
0.90,\] where \(X\) is a
binomial r.v..
The middle probability is \(0.90\) and the tail probabilities sums to
\(0.10\), with individual tails to be
\(0.05\), assuming symmetry.
Probability components:
- lower bound: \(\displaystyle
P(\overline{X} \le a) \approx 0.05 \longrightarrow a = 10\)
- upper bound: \(\displaystyle
P(\overline{X} \le b) \approx 0.95 \longrightarrow b = 23\)
Confidence Interval:
\[10 \le \overline{x} \le 23\]
\(\star\) Interpreting this interval
requires you to understand how CLT works and a basic understanding of
the frequentist interpretation of probability.
Using R:
cl <- 0.90 # confidence level
cl_tail <- 1-cl # tail probabilities
lb <- qbinom(cl_tail/2,200,16/200) # lower bound
ub <- qbinom(cl+(cl_tail/2),200,16/200) # upper bound
c(lb,ub) # confidence interval as an ordered list
## [1] 10 23
Normal Approximation
We have \(16\) spam emails and \(184\) not spam emails, both are more than
\(10\), and the sample size \(n = 200\) is considered large enough. This
rule of thumb says that we can approximate the binomial using
the normal distribution.
Normal approximation:
- The normal approximation to the binomial would yield: \[
\begin{aligned}
\overline{x} & = n\hat{p} \longleftarrow \text{ sample mean} \\
s^2 & = n\hat{p}(1-\hat{p}) \longleftarrow \text{sample variance}
\end{aligned}
\]
- Note that \[\hat{p} = \frac{\text{number
of spam emails}}{n}.\] This is known as the sample
proportion.
- So, \(\displaystyle \hat{p} =
\frac{16}{200} = 0.08\).
Confidence interval assuming normality:
\[
\overline{x} \pm z^* \cdot SE
\] where \(SE\) is called the
standard error. Here, \(\displaystyle
SE = \sqrt{n\hat{p}(1-\hat{p})}\). The term \(z^*\) is called the critical value, which
can be computed using R.
So, the interval is
\[
\begin{aligned}
\overline{x} & \pm z^* \cdot \sqrt{n\hat{p}(1-\hat{p})} \\
16 & \pm 1.645\left(\sqrt{200(0.08)(1-0.08)})\right)
\end{aligned}
\]
\[9.689 \le \overline{x} \le
22.311\]
Using R:
cl <- 0.90 # confidence level
cl_tail <- 1-cl # tail probabilities
lb <- qnorm(cl_tail/2,16,sqrt(200*0.08*(1-0.08))) # lower bound
ub <- qnorm(cl+(cl_tail/2),16,sqrt(200*0.08*(1-0.08))) # upper bound
c(lb,ub) # confidence interval as an ordered list
## [1] 9.689247 22.310753
Facebook’s categorization of user interests
Most commercial websites (e.g. social media platforms, news out-
lets, online retailers) collect a data about their users’ behaviors and
use these data to deliver targeted content, recommendations, and
ads.
To understand whether Americans think their lives line up with how
the algorithm-driven classification systems categorizes them, Pew
Research asked a representative sample of 850 American Facebook
users how accurately they feel the list of categories Facebook
has listed for them on the page of their supposed interests actually
represents them and their interests. 67% of the respondents said
that the listed categories were accurate.
Estimate the true proportion of American Facebook users who think the
Facebook categorizes their interests accurately.
Point Estimate and Standard Error
The goal of parameter estimation is to find a range of possible
values (confidence interval).
Given information
- \(\hat{p} = 0.67 \longleftarrow
\text{point estimate}\)
- \(n = 850 \longleftarrow \text{sample
size}\)
- The expected number of users who think the Facebook categorizes
their interests accurately is \(850 \times
0.67 \approx 569.5\) (569 or 570).
- There around 280.5 (280 or 281) users think the opposite.
- Let \(p\) bet he true population
proportion and \(\hat{p}\) be the
sample proportion.
The Confidence Interval
We want to find the 95% confidence interval using the formula: \[\text{point estimate} \pm 1.96 \times
\text{SE}\] where SE is the standard error.
This can be written as \[
\begin{aligned}
0.67 & \pm 1.96 \times \sqrt{\frac{0.67 (1-0.67)}{850}} \\
0.67 & \pm 1.96 \times 0.0161 \\
& \longrightarrow (0.67-0.0316,0.67+0.0316) \\
& \longrightarrow (0.6384,0.7016)
\end{aligned}
\]
Thus, the 95% interval for estimating the true \(p\) is between 0.6384 and 0.7016.
Interpretation
Which of the following is the correct interpretation of this
confidence interval? We are 95% confident that:
- 63.84% to 70.16% of American Facebook users in this sample think
Facebook categorizes their interests accurately.
- 63.84% to 70.16% of all American Facebook users think Facebook
categorizes their interests accurately
- There is a 63.84% to 70.16% chance that a randomly chosen American
Facebook user’s interests are categorized accurately.
- There is a 63.84% to 70.16% chance that 95% of American Facebook
users’ interests are categorized accurately.
What does 95% Confident Mean?
Suppose we took many samples and built a confidence interval from
each sample using the equation \[\text{point
estimate} \pm 1.96 \times \text{standard error}.\]
Then about 95% of those intervals would contain the true population
proportion (\(p\)).
Width of an interval
If we want to be more certain that we capture the population
parameter, i.e. increase our confidence level, should we use a wider
interval or a smaller interval?
\(\star\) A wider interval.
Can you see any drawbacks to using a wider interval?
\(\star\) If the interval is too
wide it may not be very informative.
Changing the Confidence Level
\[\text{point estimate} \pm z^{\star}
\times \text{SE}.\]
- In a confidence interval, \(z^{\star}
\times \text{SE}\) is called the margin of
error, and for a given sample, the margin of error changes as
the confidence level changes.
- In order to change the confidence level we need to adjust \(z^{\star}\) in the above formula.
- Commonly used confidence levels in practice are 90%, 95%, 98%, and
99%.
- For a 95% confidence interval, \(z^{\star}
= 1.96\).
- However, using the standard normal distribution, it is possible to
find the appropriate \(z^{\star}\) for
any confidence level.
95% Confidence Interval
99.7% Confidence Interval
Finding \(z^{\star}\) Exactly
Find the \(z^{\star}\) for a 92%
confidence level.
Process:
- Confidence level is 0.92.
- Lower tail of the \(Z \sim N(0,1)\)
is \(\frac{(1-0.92)}{2} = 0.04\).
- We want to find the \(z\) score
that would yield a 0.04 probability.
- Use the
qnorm() function in R.
Using R:
cl <- 0.92 # confidence level
lt <- (1-cl)/2 # lower tail probability
qnorm(lt,0,1) # computes the z star
## [1] -1.750686
