# Tip 3: Gain Confidence with Confidence Intervals

New to confidence intervals?  Here are some important things to keep in mind!

## Confidence Intervals:

• are used to estimate population parameters (commonly the process mean, standard deviation, % of defective units, or even capability indices).
• provide more meaningful information than any random sample statistic for characterizing the population.

When your 95% confidence interval for the mean is (μlow, μhigh), you can be 95% confident that the population (process) mean, μ, is between μlow and μhigh …and 5% confident that μ is not between μlow and μhigh.

When your 95% confidence interval for the standard deviation is (σlow, σhigh), you can be 95% confident that the population standard deviation, σ, is between σlow and σhigh …and 5% confident that s is not between σlow and σhigh.

MINI-TIPS:
Multiply the high estimate for σ by 6 to approximate the potential range of a normal distribution. Multiplying the sample standard deviation, σ, by 6 tells us nothing about the population.

When your 95% confidence interval for the proportion of defectives is (plow, phigh), you can be 95% confident that the process is producing at least plow x100% defectives, and potentially as many as phigh x100% defective units.

For example, suppose you inspect 100 units and find 2 defectives. The sample % defective is 2%.  But, using Minitab's 1 Proportion study, we create the 95% confidence interval for the proportion of defectives:  (0.0024, 0.070).  So this sample may be from a process that is producing a whopping 7% defective units!

TAKE-AWAY PERSPECTIVE:
There is always some RISK that even the confidence intervals will not contain the parameter of interest. That said, just imagine how risky it is to assume any sample statistic adequately characterizes the population!

## More Facts About Confidence Intervals:

• INCREASING the % confidence INCREASES the length of the interval.
• INCREASING the sample size DECREASES the length of the interval.

MINI-TIP:
Use Minitab’s ‘Power and Sample Size for Estimation’ to determine sample sizes necessary to create confidence intervals of meaningful length.

• INCREASING the sample size does not increase the likelihood that the interval will contain the population parameter. Only increasing % confidence does this.

TAKE-AWAY PERSPECTIVE:
Confidence intervals based on small samples are as credible (reliable) as those based on large samples.

• The true population parameter may be near one end of the confidence interval.

TAKE-AWAY PERSPECTIVE:
Do not assume that the parameter of interest is near the middle of the interval.

• A 95% Confidence Interval DOES NOT contain 95% of the individual values in the population.

MINI-TIP:
Use Minitab’s ‘Tolerance Intervals’ to create intervals that capture a desired percent of the individual population values, with a given degree of confidence.

In Tip 4, we'll provide a brief explanation of tolerance intervals.

To gain more perspectives on statistical analysis procedures using Minitab Statistical Software, visit Minitab's web site for course descriptions and  training opportunities, both public and on-site, at http://www.minitab.com/en-us/services/training/

### 7 Deadly Statistical Sins Even the Experts Make

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Name: Alex • Monday, January 28, 2013

Hi, folks! Your example with 100 units and find 2 defectives, where "sample may be from a process that is producing a whopping 7% defective units!" does not correspond one propotion test in MINITAB.
Test and CI for One Proportion

Test of p = 0,07 vs p not = 0,07

Exact
Sample X N Sample p 95% CI P-Value
1 2 100 0,020000 (0,002431; 0,070384) 0,036
So if P-value = 0,036 we reject the Ho.

Name: yunlong wei • Tuesday, January 29, 2013

Hi,everyone
the article above said,"For example, suppose you inspect 100 units and find 2 defectives. The sample % defective is 2%. But, using Minitab's 1 Proportion study, we create the 95% confidence interval for the proportion of defectives: (0.0024, 0.070). So this sample may be from a process that is producing a whopping 7% defective units!" do you know how to get the intervals(0.0024, 0.070)? and how to get 7% defective units?

Name: karen • Wednesday, January 30, 2013

Hello, Alex.
Thanks for your interest in the Minitab blog. It's nice to see that you are messing with Minitab to recreate the results quoted in the example.

The contradiction between the confdence interval and the hypothesis test is due to that fact that there are no confidence intervals that produce exact confidence for binomial proportions. This is due to a couple of complications: 1. the binomial distribution is not continuous, and 2. the variance of sample proportions depends on the population proportion, which, of course, in unknown. So, all confidence intervals for proportions will only be approximate. Minitab uses a method attributed to Clopper & Pearson, where the achieved level of confidence will always be at least that specified. So the confidence level in the example sited, is actually slightly higher than 95%. Since it is slightly wider than a 95% confidence interval, its upper bound is a value that is rejected by the Hypothesis test.
If you hypothesize 6.5%--a value quite near to 7%--you will find a p-value at 0.05, suggesting that 6.5% may be the population proportion. So maybe not as high as 7%, but could be greater than 6%.
I found a couple of nice articles on Wikipedia that you might enjoy, for a more detailed discussion of these principles:
http://en.wikipedia.org/wiki/Binomial_proportion_confidence_interval
http://en.wikipedia.org/wiki/Confidence_interval

hope this helps! karen

Name: karen • Wednesday, January 30, 2013

Yunlong Wei,
Thanks for your interest in Minitab blogs, and in your efforts to recreate the analaysis using Minitab.

To create the confidence interval cited in the example:
1. Select STAT > BASIC STATISTICS > 1 PROPORTION
2. Select 'SUMMARIZED DATA'
3. For 'NUMBER OF EVENTS' type 2.
4. For 'NUMBER OF TRIALS' type 100.
5. Click 'OK'

This ~95% confidence interval indicates that you can be (at least) 95% confident that the proportion of defective units in the population is at least 0.0024 = 0.24% and possibly as great as 0.0703 = 7.03%

hope this helps!
karen

Name: Alex • Thursday, January 31, 2013

Hi, Karen! Thanks for your comment. It's very interesting. But I tried the same example in R with binom.test and got the same result. This R function has confidence intervals obtained by a procedure first given in Clopper and Pearson (1934).
binom.test(2,100,0.07)

Exact binomial test

data: 2 and 100
number of successes = 2, number of trials = 100, p-value = 0.0482
alternative hypothesis: true probability of success is not equal to 0.07
95 percent confidence interval:
0.002431337 0.070383932
sample estimates:
probability of success
0.02

I hope my inspiration will not bother you!

Name: karen • Thursday, January 31, 2013

Alex,
I am glad to see that R uses the same interval construction as Minitab. Hopefully this gives you confidence that this is currently the best available method.
Given that your response says, "But, ...", it sounds like you still have a question, but its nature is not clear to me.