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CONFIDENCE INTERVALS

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- point estimate: estimate exact value
- precise
- likely to be wrong

- interval estimate: range of values
- less precise
- less likely to be wrong

1. Range of values

2. Level of confidence

- No hypothesis is directly tested

- Purpose: provide a range of values containing the population mean
- Design: any design where a mean is calculated
- Assumptions: same as for z-test

- Start with your sample mean as the middle of the interval.
- Establish upper and lower limits depending on how confident you want to be.

lower limit

upper limit

M

A sample of 25 students took an aptitude test with a population standard deviation of 40. The mean of the 25 students was 500. Compute the 95% confidence interval for the mean.

STEP 1: Compute the standard error.

STEP 2: Find the z score for your confidence level.

For 95% confidence, use z = 1.96

For 99% confidence, use z = 2.58

z = 1.96

STEP 3: Compute the lower and upper limits.

The 95% confidence interval for the mean aptitude test score was 484.32 - 515.68.

- Purpose: provide a range of values containing the population mean
- Design: any design where a mean is calculated
- Assumptions: same as for single sample t-test

64 research participants were timed on a motor task. Their mean time was 20 sec, with s = 4.00. Compute the 99% confidence interval for the mean.

STEP 1: Compute the standard error.

- STEP 2: Find the t score for your confidence level. Look up two-tailed t in table, using df = N-1, and 1- level of confidence for a.
- df = 64-1=63
- a = 1-.99=.01
- two-tailed t = 2.660

STEP 3: Compute the lower and upper limits.

The 99% confidence interval for the mean time to complete the task was 20 sec +/- 1.33.

- Purpose: provide a range of values containing the population proportion
- Design: any design where a proportion or percent is calculated
- Assumptions:
- representative sample
- independent observations

A sample of 500 registered voters rated how well the President is doing his job. The proportion who gave him a “good” rating was .52, or 52%. Compute the 95% confidence interval for the proportion.

STEP 1: Compute the standard error.

- STEP 2: Find the z score for your confidence level.
- For 95% confidence, use z = 1.96
- For 99% confidence, use z = 2.58
- z = 1.96

STEP 3: Compute the lower and upper limits.

The 95% confidence interval for the proportion of “good” ratings was .52 +/- .04.

- A wider interval is less precise and therefore less informative.
- The interval will be narrower with
- a larger sample
- lower variability
- a lower level of confidence

- The media often report poll results and include margin of error. This is the +/- part, usually in fine print.
- Add and subtract the margin of error to get the confidence interval

- Can be used as an indirect test of significance
- Construct confidence interval around sample statistic using margin of error
- If interval includes Ho value, difference is not significant