AP Statistics t Intervals and t Tests for a Mean Lesson (Free PDF)
This inference one mean lesson for AP Statistics covers t intervals and t tests for a mean (Common Core AP STAT 7.1, AP STAT 7.2, AP STAT 7.3, AP STAT 7.4, AP STAT 7.5, HSS-IC.B.4), with a free PDF to print.
When the population standard deviation is unknown, inference for a mean uses the t distribution with n – 1 degrees of freedom. A one-sample t interval is x-bar plus or minus t* s/sqrt(n); a t test uses t = (x-bar – mu0)/(s/sqrt(n)). Conditions: random sample, 10 percent, and a normal population or large sample. For matched pairs, analyze the differences.
AP Statistics · Common Core State Standards: AP STAT 7.1, AP STAT 7.2, AP STAT 7.3, AP STAT 7.4, AP STAT 7.5, HSS-IC.B.4
Key facts in this inference one mean lesson
- t distribution: bell-shaped, centered at 0, heavier tails; df = n – 1.
- Interval: x-bar +/- t* s/sqrt(n); test: t = (x-bar – mu0)/(s/sqrt(n)).
- Conditions: random, 10% condition, and Normal/Large Sample (population normal, n >= 30, or no strong skew or outliers).
- As df grows, t* approaches z*.
- Matched pairs: take differences and do a one-sample t procedure.
Why use the t distribution?
When we estimate a population mean \(\mu\) and do not know the population standard deviation \(\sigma\), we use the sample standard deviation \(s\). That extra uncertainty is handled by the t distribution, which is bell-shaped, centered at 0, but with heavier tails than the normal. It depends on the degrees of freedom \(df=n-1\).
The standard error of the mean is \(SE_{\overline{x}}=\frac{s}{\sqrt{n}}\). As \(df\) grows, the t critical values shrink toward the normal values.
How do you build and check a one-sample t interval?
\[\overline{x}\pm t^{*}\frac{s}{\sqrt{n}}\]
- Random: a random sample or randomized experiment.
- 10%: \(n\) is less than 10% of the population when sampling without replacement.
- Normal/Large Sample: the population is approximately normal, or \(n\ge 30\), or the sample data show no strong skew or outliers.
Example: a sample of 12 batteries has mean life 40.36 hours and \(s=2.20\) hours. The dotplot is roughly symmetric, so the Normal condition is reasonable.
\(SE=\frac{2.20}{\sqrt{12}}=0.634\), \(df=11,\ t^{*}=2.201\), \(ME=2.201(0.634)=1.395\). The interval is (38.96, 41.75).
We are 95% confident that the interval from 38.96 to 41.75 hours captures the true mean battery life of all batteries of this type.
How do you carry out a one-sample t test?
For \(H_0:\mu=\mu_0\), the test statistic is:
\[t=\frac{\overline{x}-\mu_0}{s/\sqrt{n}},\qquad df=n-1\]
The p-value is the area in the tail(s) of the t distribution with \(df\) degrees of freedom. Example: if \(t=2.35\) with \(df=11\) and \(H_a:\mu>\mu_0\), the p-value is \(0.0192\).
Because 0.0192 is less than 0.05, we reject \(H_0\) and conclude that there is convincing evidence that the mean is greater than \(\mu_0\).
What about matched pairs?
When each individual is measured twice (before and after), compute the differences and treat them as one sample. Use a one-sample t procedure on the differences, with \(H_0:\mu_d=0\). Check Normality using the differences, not the original data.
Inference one mean lesson: questions and answers
Why is the t distribution used instead of the normal?
Using s instead of sigma adds variability, so the standardized statistic has heavier tails than the normal. The t distribution accounts for that, especially for small samples.
What are degrees of freedom?
For a one-sample t procedure, df = n – 1. They tell which t curve to use; larger df gives a curve closer to the normal curve.
How do I check the Normal condition for a small sample?
Plot the data. If the graph shows no strong skewness and no outliers, a t procedure is reasonable. For very small samples, you need a population that is approximately normal.
When do I use a matched pairs t procedure?
When each subject is measured twice or two similar subjects are paired. Compute the differences and run a one-sample t procedure on them.
Can I use a t procedure with a skewed population?
If the sample is large (n at least 30), the central limit theorem makes the sampling distribution of x-bar approximately normal, so yes. With small n and clear skew or outliers, avoid t procedures.
How to use this inference one mean lesson
Read the definitions first, then follow the worked method line by line and redo the example on paper without looking. When every step makes sense, move on to the worksheet, then check yourself with the quiz.
Download the inference one mean lesson as a PDF with the button below and keep it with your notes.
Download the inference one mean lesson as a PDF with the button below and keep it with your notes.
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