Confidence Intervals

The interval moves; the parameter does not

Slides 15 through 17 are the most important sequence in the deck, and they are easy to flip past because each looks almost identical to the last.

In all three, the true mean μ is drawn as a fixed vertical line. What changes is which sample you happened to draw:

• A sample from the left tail → the interval sits low, but still covers μ.
• A sample from the right tail → the interval sits high, but still covers μ.
• A sample from the far right tail → the interval sits too high and misses μ.

Then the question: how often will this happen? Five percent of the time, by construction.

This is the single most misread idea in applied statistics. A 95% confidence interval does not mean there is a 95% chance the true value is inside your interval. Your interval either contains it or it does not — you just cannot tell which. The 95% is a property of the procedure: repeat the study many times, and 95% of the intervals you build this way will capture the truth.

Read the three slides as three draws from the same machine, not as three different situations.

The Central Limit Theorem is load-bearing, not a footnote

Slide 8 is presented as an “aside,” but nothing after it works without it.

A standard error is a single number describing spread. To turn a spread into an interval you also need to know the shape of the sampling distribution — how far out you must go to capture 95% of it. The CLT supplies that shape for free, and crucially it does so regardless of what the population looks like. The population can be skewed, lumpy, bimodal; the distribution of the sample statistic still converges to normal.

That is why the slide’s parenthetical is not a joke: otherwise we would not have inferential statistics. Without a known shape, a standard error would be uninterpretable.

Everything is taught twice, on purpose

This deck has a strict two-column structure, and it is the same structure as the road map:

1. Establish the concept on the mean, where you already have intuition from your first statistics course.
2. Re-establish it on the slope, where the arithmetic differs but the logic is identical.

Sampling distribution of the mean → sampling distribution of the slope. Standard error of the mean → standard error of the slope. Confidence interval of the mean → confidence interval of the slope.

If a slope concept feels slippery, find the matching mean slide and reason there first. The transfer is exact; only the formula changes.

Where “significance” is about to come from

Slide 19 plots the confidence interval of the slope against a dashed vertical line at zero. The deck does not say the word yet, but that picture is the whole of hypothesis testing:

• If the interval excludes zero, you can rule out “no relationship” at that confidence level. The result is statistically significant.
• If the interval contains zero, you cannot.

That is why the standard error matters so much in practice. It sets the width of the interval, which decides whether zero falls inside. Chain the whole course together and you get:

Every lecture from p-02 forward has been building one link of that chain.

Answering the deck’s own third question

“We care about the sampling variance of which statistic in regression?”

The slope, b₁. Not of Y, not of X, not of the residual. Those all feed into the calculation, but the thing whose sampling distribution we actually care about — the thing we build an interval around and test — is the estimated coefficient.

This is worth saying out loud because the arithmetic spends most of its time on the variance of Y and the variance of X, which makes it easy to lose track of what the answer is about.

A note on the slides’ notation. Slides 18 and 19 both write the interval with a greater-than sign on the right — b₁ − t · SE < β₁ > b₁ + t · SE. That should be a second less-than sign; as written it says β₁ is larger than both endpoints, the opposite of the intended meaning. Read both as < … <.

How to read the slides

1. What is a sampling distribution? The distribution of a statistic across repeated samples from the same population — not the distribution of your data. A dot in it is one study, not one observation.
2. What is the relationship between the sampling distribution and the standard error? The standard error is the standard deviation of the sampling distribution. It measures the typical distance from an estimate to the truth.
3. We care about the sampling variance of which statistic in regression? The slope, b₁. Everything else is an input to computing it.
4. What role does the standard error play in the confidence interval? It sets the width. The interval is the estimate plus or minus t standard errors, so a smaller standard error means a tighter interval and a more informative result.

Key takeaways

• A sampling distribution is made of statistics, not observations. One dot is one whole study.
• The standard error is the standard deviation of the sampling distribution — the average distance from an estimate to the truth.
• The Central Limit Theorem supplies the shape that turns a standard error into an interval, and it works regardless of the population’s shape.
• The 95% belongs to the procedure. Across many samples, 95% of the intervals built this way capture the parameter. Your particular interval either does or does not.
• The parameter is fixed; the interval is what moves. Slides 15–17 are three draws from one machine.
• Everything is taught on the mean first, then transferred to the slope. When the slope version is confusing, go back to the mean version.
• An interval that excludes zero is a significant result — this is where the course goes next.