Notes
Normal distributions
- Normal data are symmetric and bell-shaped. Empirical Rule: about 68%, 95% and 99.7% fall within 1, 2 and 3 standard deviations of the mean.
- z-score $\displaystyle = \frac{x - \mu}{\sigma}$. It measures distance from the mean in standard deviations and lets you compare different distributions.
Collecting data
- A simple random sample gives every member an equal chance. Convenience and voluntary response samples are biased.
- Observational studies show association only. Experiments with random assignment and a control group can show cause and effect.
- Random sampling lets you generalize. Random assignment supports cause and effect.
Inference and simulation
- Margin of error $\displaystyle \approx \frac{1}{\sqrt{n}}$. The interval is $\hat{p} \pm ME$. Larger samples give smaller margins.
- Simulations assign random digits to match probabilities, then repeat many trials.
Topics in this unit
Each topic has its own short notes and practice.
Practice
68% of values are within 1 standard deviation: $100 \pm 15$.
$\displaystyle z = \frac{172 - 165}{7} = 1$.
People with strong opinions are more likely to respond, and a large biased sample is still biased.
Random assignment balances other factors across groups, so differences can be attributed to the treatment.
$\displaystyle \frac{1}{\sqrt{2500}} = \frac{1}{50} = 0.02$.
Two of ten digits is 20%.
Free response
Write your answer first, then open the scoring guide.
Free-response question
A poll surveys a random sample of 600 voters about a ballot measure. 312 of them say they support it.
(a) Find the sample proportion in favor.
Show the answer and scoring guide
$\displaystyle \hat{p} = \frac{312}{600} = 0.52$, or 52%.
Check off the points you earned:
(b) Use the margin of error $\frac{1}{\sqrt{n}}$ to find an interval for the true proportion.
Show the answer and scoring guide
$\displaystyle ME = \frac{1}{\sqrt{600}} \approx 0.041$. The interval is $0.52 \pm 0.041$, or about 0.479 to 0.561.
Check off the points you earned:
(c) Can the pollster conclude that a majority of all voters support the measure? Explain.
Show the answer and scoring guide
No. The interval includes values below 0.50, so the true proportion could be less than a majority.
Check off the points you earned: