- How to Find a Confidence Interval for a Sample
- How to Find a Confidence Interval for a Proportion: Overview
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When you don’t know anything about a population’s behavior (i.e. you’re just looking at data for a sample), you need to use the t-distribution to find the confidence interval. That’s the vast majority of cases: you usually don’t know population parameters, otherwise you wouldn’t be looking at statistics!
The confidence interval tells you how confident you are in your results. With any survey or experiment, you’re never 100% sure that your results could be repeated. If you’re 95% sure, or 98% sure, that’s usually considered “good enough” in statistics. That percentage of sureness is the confidence interval.
Confidence Interval For a Sample: Steps
A group of 10 foot surgery patients had a mean weight of 240 pounds. The sample standard deviation was 25 pounds. Find a confidence interval for a sample for the true mean weight of all foot surgery patients. Find a 95% CI.
Step 2: Subtract the confidence level from 1, then divide by two.
(1 – .95) / 2 = .025
Step 3: Look up your answers to step 1 and 2 in the t-distribution table. For 9 degrees of freedom (df) and α = 0.025, my result is 2.262.
|df||α = 0.1||0.05||0.025||0.01||0.005||0.001||0.0005|
Step 4: Divide your sample standard deviation by the square root of your sample size.
25 / √(10) = 7.90569415
Step 5: Multiply step 3 by step 4.
2.262 × 7.90569415 = 17.8826802
Step 6: For the lower end of the range, subtract step 5 from the sample mean.
240 – 17.8826802 = 222.117
Step 7: For the upper end of the range, add step 5 to the sample mean.
240 + 17.8826802 = 257.883
That’s how to find the confidence interval for a sample!
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When we talk about a confidence interval (CI), we’re dealing with data. For example, let’s say the manager for that job you applied for told you he would get back with you in a “couple of days.” A couple of days could mean two. Or three. Or there might be a paperwork backlog and it could be a week. It definitely doesn’t mean in an hour. So your CI would probably be between 2 and 4 days.
Perhaps the trickiest part of CIs is recognizing the various parts needed for the formula, like z a/2. This section breaks everything down into simple steps and shows you how to find a confidence interval for population proportions.
How to Find a Confidence Interval for a Proportion: Steps
Question: 510 people applied to the Bachelor’s in Elementary Education program at Florida State College. Of those applicants, 57 were men. Find the 90% CI of the true proportion of men who applied to the program.
Step 1: Read the question carefully and figure out the following variables:
- α : subtract the given CI from 1.
- z α/2: divide α by 2, then look up that area in the z-table. Not sure how to read a z table? See this article, area to the right of a z-score for an explanation on how to read the z-table.
.1/2=.0500. The closest z-value to an area of .0500 is 0.13
- : Divide the proportion given (i.e. the smaller number)by the sample size.
- : Subtract from 1.
1-0.112 = 0.888
Step 2: Multiply by .
0.112 x 0.888 = 0.099456
Step 3: Divide step 2 by the sample size.
0.099456 / 510 = 0.000195011765
Step 4:Take the square root of step 3:
sqrt(0.000195011765) = 0.0139646613
Step 5: Multiply step 4 by z a/2:
0.0139646613 x 0.13 = 0.0182
Step 6:: For the lower percentage, subtract step 5 from .
0.112-0.0182 = 0.0938 = 9.38%
Step 7:For the upper percentage, add step 5 to .
0.112 + 0.0182 = 0.1302 = 13.02%
That’s how to find a confidence interval!
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