rate = r
πINTRODUCTORY STATISTICS STUDY GUIDE
Confidence Intervals
Build and interpret intervals for proportions and means while checking the required conditions.
8 key ideas8 flashcards5 practice questions
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rate = 2ΚΈr
use the measured ratio
Key ideas to know
Start with the relationships between ideas. Then close the notes and explain each one from memory.
- 01
A confidence interval estimates an unknown population parameter.
- 02
The margin of error combines a critical value and standard error.
- 03
Higher confidence creates a wider interval.
- 04
Larger samples reduce standard error.
- 05
Confidence describes the long-run success rate of the method.
- 06
A one-proportion interval uses an estimated proportion plus or minus a confidence multiplier times its standard error.
- 07
A one-mean t interval uses sample mean, sample standard deviation, sample size, and t degrees of freedom.
- 08
Systematic bias is not repaired by a larger sample or a narrower interval.
Two ideas worth correcting now
NOT QUITE
βA 95 percent interval has a 95 percent probability of containing the fixed parameter after calculation
USE THIS INSTEAD
The confidence rate describes the method across repeated samples.
NOT QUITE
βMore observations remove every source of error
USE THIS INSTEAD
A larger sample reduces random error but not systematic bias.
Flashcards
Answer before opening each card. The effort to retrieve is part of the learning.
1What is the general confidence-interval form?Show answer οΌ
Estimate Β± margin of error.
2What happens to interval width when sample size increases?Show answer οΌ
The interval becomes narrower, all else equal.
3What does 95% confidence mean?Show answer οΌ
About 95% of intervals from repeated samples using the method would capture the true parameter.
4When is a t interval used for a mean?Show answer οΌ
When the population standard deviation is unknown.
5Does a confidence interval prove causation?Show answer οΌ
No. Causation depends on study design, not the interval alone.
6What conditions are checked for a one-proportion interval?Show answer οΌ
Randomness or a defensible sampling process, independence, and enough expected successes and failures.
7What determines t degrees of freedom for one mean?Show answer οΌ
Sample size minus one.
8Does a narrow interval repair biased sampling?Show answer οΌ
No. It can be precisely wrong.
Explain it in your own words
Use the answer as a check after you have written or spoken your response.
01Why is '95% of the population lies in the interval' incorrect?
The interval estimates a parameter, not the spread of individual observations.
02What tradeoff comes with greater confidence?
The interval becomes wider and less precise.
03Why check independence?
Standard-error formulas assume observations do not provide redundant information.
04Why does quadrupling sample size roughly halve margin of error?
Standard error usually contains the square root of sample size in the denominator.
05What happens when the data are strongly skewed and the mean sample is small?
The t procedure may be unreliable, so the sampling setting and outliers require care.
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