Confidence Intervals I



Megan Ayers

Math 141 | Spring 2026
Monday, Week 7

Announcements/reminders

  • Homework 5 due tonight

  • Homework 6 is posted, but not due until 3/20 (last day before spring break)

    • Today’s content relevant to starting Ex. 4, 5.

Today’s Goals

  • Introduce confidence intervals as a method for estimating a parameter

  • Use bootstrapping as a means of creating confidence intervals

Confidence Intervals

Point Estimates

  • To estimate a population parameter, we can use a sample statistic.
    • Ex: You’re hosting a pizza party for 200 people, and need to know what proportion \(p\) of vegetarian pizza to order.
    • Idea: Ask a (random) sample of attendees about their pizza preference and use the proportion \(\widehat{p}\) in the sample as an estimate for the total proportion \(p\).
  • Your sample statistic, \(\widehat{p}\), is a good guess for the population parameter.
    • Terminology: We sometimes call our sample statistic a point estimate.

Point Estimates vs. Interval Estimates

  • A sample statistic is a good guess for the population parameter, but not the whole story
    • Q: After polling pizza party attendees, you find \(\widehat{p} = 0.33\). What factor(s) determine your (un)certainty in this estimate?
  • It may be preferable to estimate the proportion using a range of values, with smaller intervals corresponding to precision.

    • With just \(n = 9\) people, you might give a range of \(0.03\) to \(0.63\) for \(p\).

    • But with \(n = 48\), you might instead give the range \(0.20\) to \(0.46\).

  • We call these ranges interval estimates.

Confidence Interval Estimates

A confidence interval estimate for a parameter usually takes the form \[ \textrm{Statistic }\pm \textrm{ Margin of Error (ME)} \]

The confidence interval gives a range of plausible values for the parameter.

  • e.g., when sampling pizza preferences with \(n=48\), we estimate \(p\) using the interval

\[ 0.20 \textrm{ to } 0.46 \qquad \textrm{ or } \qquad \underbrace{0.33}_{\text{Statistic ($\widehat{p}$)}} \pm \ \ \ \underbrace{0.13}_{\text{ME}} \]

The Margin of Error determines the width of the interval (\(2*\text{Margin of Error}\))

  • e.g., in our pizza interval, the width is: \[ 0.46 - 0.20 = 0.26 = 2 * \underbrace{0.13}_{\text{ME}} \]

Q: How should we choose our Margin of Error? (think conceptually)

Confidence Intervals using the Sampling Distribution

  • Goal: Figure out a reasonable Margin of Error

  • In our example, suppose \(p = 0.25\) (i.e., 25% of all party attendees prefer vegetarian)

  • For approximately bell-shaped sampling distributions, 95% of all sample statistics are within 2 SE of the parameter.

  • This also means that for 95% of all samples, the parameter will be within a distance of 2 SE of the sample statistic (every sample in the gray region)

Confidence Intervals

Idea: Build an interval centered at the sample statistic, with a margin of error of \(2\cdot SE\):

\[ \widehat{p} \pm 2\cdot \textrm{SE} \]

  • This interval will contain the parameter \(p\) in 95% of samples!

  • The interval for our sample was \(0.33 \pm 0.125\), which does contain the parameter \(p\)

Interval Estimates

Idea: Build an interval centered at the sample statistic, with a margin of error of \(2\cdot SE\):

\[ \widehat{p} \pm 2\cdot \textrm{SE} \]

  • This interval will contain the parameter \(p\) in 95% of samples!

  • Samples with \(\widehat{p}\) in the gray region have intervals that also contain the parameter \(p\)

Interval Estimates

Idea: Build an interval centered at the sample statistic, with a margin of error of \(2\cdot SE\):

\[ \widehat{p} \pm 2\cdot \textrm{SE} \]

  • This interval will contain the parameter \(p\) in 95% of samples!

  • Samples with \(\widehat{p}\) outside the gray region have intervals that don’t contain \(p\)

Interval Estimates

Idea: Build an interval centered at the sample statistic, with a margin of error of \(2\cdot SE\):

\[ \widehat{p} \pm 2\cdot \textrm{SE} \]

  • This interval will contain the parameter \(p\) in 95% of samples!

  • But 95% of all samples will have intervals that do contain \(p\)

Confidence levels

  • On the previous slide, 95% of confidence intervals contained the true parameter, \(p\).

  • Thus, we call each interval estimate a 95% confidence interval.

  • Here, 95% is our confidence level: The success rate for our estimation technique.

    • e.g., The pizza interval \(0.33 \pm 0.13\) has confidence level of 95%

  • The confidence level corresponds to the percentage of samples that would yield a corresponding confidence interval containing the true value of the parameter.

Confidence levels

  • Confidence Level: the percentage of samples that would yield a corresponding confidence interval containing the true value of the parameter.

  • For example, were we to:

    • Repeatedly draw samples from the population, and
    • Create 95% confidence intervals in each sample…
    • 95% of those confidence intervals would contain the true parameter

Recap: Confidence Intervals

  • Confidence intervals consist of 1. an interval estimate and 2. a confidence level.

  • In our pizza party sample, we estimated the true proportion of vegetarian pizza-eaters was between \(0.20\) and \(0.46\), with \(95\%\) confidence.

  • What does “95% confidence” mean?

    • It’s the success rate of the process
    • For 95% of all samples, the interval we construct will actually contain the parameter.

  • Worth emphasizing: it’s the success rate of The Process, NOT the specific interval you calculated in your sample.

    • The parameter is either in your interval, or it’s not – there’s no success rate there!

Think-Pair-Share

Q: What’s the difference between these two interpretations of a 95% confidence interval?

  • For 95% of all samples, the interval we construct will actually contain the parameter.

  • For a given interval, there is a 95% chance that the parameter will fall in the interval.

Think-Pair-Share (Answer)

Q: What’s the difference between these two interpretations of a 95% confidence interval?

  • For 95% of all samples, the interval we construct will actually contain the parameter.

  • For a given interval, there is a 95% chance that the parameter will fall in the interval.


A:

  • The first one is accurate! 95% confidence refers to the accuracy of the process

  • The second one is wrong, an easy mistake to make!! It confuses probability of the process with a deterministic outcome. Our sample statistic “moves” with sampling variability - the parameter does not.

Interval Width and Sample Size (\(n\))

Idea: Build a 95% confidence interval centered at the sample statistic, with a margin of error of \(2\cdot SE\): \[ \widehat{p} \pm \underbrace{2\cdot \textrm{SE}}_{\text{Margin of Error}} \]

  • This interval’s width is determined by the Standard Error (SE).

  • Reminder: The SE is the standard deviation of the sampling distribution.

  • Q: What do we know about the SE as the sample size (\(n\)) increases?

  • Q: What does this imply about the interval’s width as the sample size (\(n\)) increases?

    • The SE gets smaller as \(n\) increases!
    • Our interval becomes narrower as \(n\) increases!

Interval Width and Confidence Level

Idea: Build an 95% confidence interval centered at the sample statistic, with a margin of error of \(2\cdot SE\): \[ \widehat{p} \pm \underbrace{2\cdot \textrm{SE}}_{\text{Margin of Error}} \]

  • Chose \(\text{Margin of Error} = 2*SE\) because 95% of sample statistics fall within \(2*SE\) of the parameter (in sampling distribution).

  • Fun Fact: 90% of sample statistics fall within \(1.64*SE\) of the parameter.

    • Q: What would happen if we did the following interval? \[ \widehat{p} \pm 1.64*\textrm{SE} \]
    • We would get a 90% confidence interval!
    • This interval is narrower, but has a lower “success” rate! For 90% of all samples, the interval will contain the parameter.

Interval Width and Confidence Level

  • Takeaway: The confidence level also determines the width of the interval!

    • higher confidence (e.g., 95%) means a larger margin of error (wider interval)

    • lower confidence (e.g., 90%) means a smaller margin of error (narrower interval)

Problems(?) with Confidence Intervals

Let’s say we’re working with 95% confidence intervals

  • Problem 1: We only have 1 sample, and we don’t know if it belongs to the 95% of “good” samples, or the 5% of “bad” ones
    • Consolation: If I go through life constructing 95% confidence intervals, I will be right about 95% of the time
    • That’s not bad!


  • Problem 2: To make a confidence interval, we need the sampling distribution in order to compute the standard error. But in practice, we (often) don’t have direct access to this.
    • Solution: approximate the sampling distribution via bootstrapping!

Bootstrap Confidence Intervals

Sleep for Reed Students

Suppose we wish to estimate how many hours Reed students sleep, on average with a sample of 50 students, who we surveys on hours of sleep.

head(sleep, 4)
  Student    Hours
1       1 7.476953
2       2 7.717175
3       3 8.281295
4       4 7.117568
sleep %>% summarize(MeanSleep = round(mean(Hours), 3))
  MeanSleep
1     7.046
  • Is the true average hours of sleep at Reed is 7.046?
    • Surely not! This is just one sample of size 50
  • Let’s create a confidence interval for the true average hours
    • We can use the bootstrap distribution to estimate the SE needed for the interval

Bootstrap Average Sleep

Create the bootstrap samples:

  • Q: What is each argument of rep_sample_n() doing?

  • Each bootstrap sample consists of 50 observations sampled with replacement from the original sample (size = 50)

  • We have a total of 10,000 bootstrap samples (reps = 10000)

bootstrap_samples <- sleep %>% 
  rep_sample_n(size = 50, replace = TRUE, 
               reps = 10000)
bootstrap_samples
# A tibble: 500,000 × 3
# Groups:   replicate [10,000]
   replicate Student Hours
       <int>   <int> <dbl>
 1         1      28  4.40
 2         1      12  6.45
 3         1       7  7.71
 4         1       4  7.12
 5         1       9  7.12
 6         1       1  7.48
 7         1      27  7.26
 8         1      39  6.38
 9         1      37  3.74
10         1      44  7.98
# ℹ 499,990 more rows

Bootstrap Average Sleep

Compute bootstrap statistics: (Mean of each bootstrap sample)

bootstrap_stats <- bootstrap_samples %>% 
  group_by(replicate) %>% 
  summarize(x_bar = mean(Hours))
bootstrap_stats
# A tibble: 10,000 × 2
   replicate x_bar
       <int> <dbl>
 1         1  7.05
 2         2  6.97
 3         3  6.94
 4         4  7.31
 5         5  7.36
 6         6  7.11
 7         7  6.93
 8         8  7.10
 9         9  7.06
10        10  6.76
# ℹ 9,990 more rows
  • We now have 10,000 sample means based on the bootstrap samples, and can assess their variability

Bootstrap Average Sleep

Graph the bootstrap distribution:

  • Use the bootstrap distribution to estimate the standard error:
bootstrap_stats %>% summarize(SE = sd(x_bar))
# A tibble: 1 × 1
     SE
  <dbl>
1 0.165

Confidence Interval for Average Sleep

  • Our sample average sleep was \(\bar x = 7.046\).

  • Based on the bootstrap distribution, this statistic has Standard Error=0.165.

  • Our 95% confidence interval for the true average hours of sleep for Reed students is: \[ {7.046} \pm 2 \cdot \color{blue}{0.165} \]

    • Our best guess for average nightly sleep is that it’s between 6.716 and 7.376. This method has a success rate of \(95\%\).

Next time:

  • More on changing confidence levels

  • What to do with distributions that aren’t bell-shaped (percentile method)

  • Confidence interval misconceptions