confidence and confidence intervals

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© 2014 Institute of Industrial Engineers 1-4-1

Chapter 1-4

Confidence and Confidence Intervals

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Inferential Statistics

Process of sampling characteristics from larger populations, summarizing those characteristics, and drawing conclusions or making predictions from the summary information.

http://content.comicskingdom.net/Zits/Zits.20100228_large.gif

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Sampling Errors

The probability that the inference will be correct is referred to as the degree of confidence with which the inference can be stated.

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Sampling Errors

Type I Error

• Stating that the results of sampling are unacceptable when in reality they are.

• Symbol is a.

Type I Error

• Probability of rejecting what should be accepted.

• Level of Significance

What is 1 – a?

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Sampling Errors

Type II Error

• Stating that the results of sampling are acceptable when in realty they are not.

• Symbol is b.

Type II Error

• Probability of Accepting what should be Rejected.

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Confidence Interval

Range of values that has a specified likelihood of including the true value of a population parameter. It is calculated from sample calculations of the parameters.

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Confidence Intervals

• A confidence interval gives an estimated range of values which is likely to include an unknown population parameter, the estimated range being calculated from a given set of sample data.

• If independent samples are taken repeatedly from the same population, and a confidence interval calculated for each sample, then a certain percentage (confidence level) of the intervals will include the unknown population parameter. Confidence intervals are usually calculated so that this percentage is 95%, but we can produce 90%, 99%, 99.9% (or whatever) confidence intervals for the unknown parameter.

• The width of the confidence interval gives us some idea about how uncertain we are about the unknown parameter (see precision). A very wide interval may indicate that more data should be collected before anything very definite can be said about the parameter.

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Comparison

Point Estimates Interval Estimates

Dear Abby,

You wrote in your column that a woman is pregnant for 266 days. Who said so? I carried my baby for 10 months and 5 days. My husband is in the Navy and it could not have been conceived any other time because I only saw him once for an hour, and I didn’t see him again until the day after the baby was born. I don’t drink or run around, and there is no way the baby isn’t his, so please print a retraction about the 266-day carrying time because I am in a lot of trouble!

-San Diego Reader

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General Format

Point Estimate +

(Confidence)(Variability)(Adjustment)

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Large and Small Samples (Means)

Large

– Greater than 30

– Use z values for confidence values (Table C)

Small

– Less than 30

– Use t values for confidence values (Table E)

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Example

• For large samples the standard normal curve is used to specify the confidence. The value used is the z value.

• What z value corresponds to an a of .05 (95 percent confidence?)

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Solution

• Area outside of interval is symmetrical.

• Each tail includes a/2 or .05/2 = .025

• Table C indicates that the z value corresponding to a tail value of .025 is 1.96.

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Example

What z value corresponds to an a of .01 (99 percent confidence?)

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Example

• The student t distribution is used with small samples.

• We must know a as well as the degrees of freedom (df).

• What t value corresponds to 95 percent confidence and 15 degrees of freedom?

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Solution

The area outside a confidence interval is symmetrically distributed on each side, so the two tailed a line is used. For a = .05 and df = 15 the table shows t = 2.131.

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Example

What t value corresponds to 90 percent confidence with 22 degrees of freedom?

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General Format

Point Estimate +

(Confidence)(Variability)(Adjustment)

On the following pages are some representative

confidence interval formulas. The ones presented are not

to be considered a comprehensive listing.

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Confidence Intervals for Means Large Samples (n > 30)

n

szx

))(( 2/a

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Example

The average weight of plating used on a sample of 100 contacts was 75mg with a sample standard deviation of 6. What is the 90 percent confidence interval for the true value of the mean?

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Confidence Intervals for Means Small Samples (n < 30)

1

))(( 2/

ndf

n

stx a

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Example

• 10 samples of a component are inspected and the tensile strength is measured. The average is 18 and the standard deviation is 1.5

• What are 99 percent confidence limits?

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Question

Why don’t EXCEL and SPCXL give the same answers? (Applies to Office 2007 only.)

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Confidence Limits for Proportions – Large

Samples

n

ppzp

)1()( 2/

a

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Example

• 20,000 light bulbs were received.

• 400 were inspected

• 12 were defective

• 99 percent confidence interval

• p = 12/400 = .03

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Confidence Limits for Proportions Small Samples

1

)1()( 2/

ndf

n

pptp a

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Example

• Average proportion was .88888.

• Sample size was 9.

• Determine 99 percent confidence interval.

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Solution

n

pptp

)1)(()( 2/

a

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Standard Deviation

1 -n df

11

2/2/1

22

aa

s

ns

ns

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Example

A sample of 30 had a sample standard deviation of 14. Construct a 95% confidence interval for the true population value.

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Sample Size Estimation Variables Data

Relative Error

Where E is the maximum allowable error expressed as a decimal (%)

Absolute Error

Where E is the amount of error expressed as a measurement

22

22

2/

)()(

)()(

xE

szn a

Half Interval Width is E times the Average for variables data.

2

22

2/

)(

)()(

E

szn a

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Sample Size Proportions


2

2/)1(

E

zppn a

Half Interval Width is E for proportion data. It is always expressed as a proportion.

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Example

A young industrial engineer wants set a stopwatch time standard correctly. Based upon a preliminary study of 10 cycles she has determined that for one of the defined work elements the average time is 2.4 minutes and the sample standard deviation is 3.0. If she wants to be 95% confident that the standard she sets is accurate to within + 5% how many total repetitions much she time?

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SPCXL®

)x(E)( Width Interval Half

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Example

• A marketing survey will be used to determine the likelihood of customers buying an 8 bladed razor. The client would like to be 95 percent confident that the results of the survey are accurate to within 4% of the believed 75% “like” rate.

• How big should the sample size be?


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Practice Problems

1. Measurements of 10 sample inside diameters on washers resulted in a sample mean of .43 centimeters and a sample standard deviation of .065 centimeters. Between what limits can one be 95 percent confidence that the true inside diameter is found?

2. Measurements of 100 sample outside diameters on washers resulted in a sample mean of 4.21 centimeters and a sample standard deviation of .134 centimeters. Between what limits can one be 99 percent confident that the true outside diameter is found?

3. Measurements of 15 detergent bottles showed the average fill weight to be 15.86 ounces with a standard deviation of .55 ounces. Between what limits can one be 99 percent confident that the true fill weight is found?

4. Measurements of a sample of 28 tires indicated an average tread life of 42,500 miles with a standard deviation of 2,150 miles. Between what limits can it be said with 90 percent confidence that the true average tread life lies?

5. Inspection of 32 samples from a manufacturer indicated that 2 did not meet specifications. Between what limits can it be said with 95 percent confidence that the true proportion of defective material lies?

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Practice Problems

6. In the past month, 25 out of 150 samples have been rejected at final inspection. Between what limits can it be said with 99 percent confidence that the true proportion of rejected material lies?

7. The average number of defects produced last year was 16 per shift per day with a standard deviation of 2.5.

a) Determine 95 percent confidence limits for the mean and standard deviation based on a sample of 100 days.

b) Determine 95 percent confidence limits for the mean and standard deviation based on a sample of 10 days.

c) What is the difference between the two samples?

8. Five computer chips were produced. As part of the inspection procedure, the average time required to perform a very lengthy calculation was measured. The average time was 2.2 seconds with a standard deviation of .06 seconds. Within what limits can it be said, with 99 percent confidence, that the true processing time lies?

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Practice Problems

9. Five hundred thousand potato chips were produced. As part of the inspection procedure, a sample of 200 was checked for percent of weight as oil. The sample check indicated an average of 28 percent with a sample standard deviation of 6 percent. Calculate a 90 percent confidence interval for this statistic. What would have happened to this interval if a sample four times as big (800) had been taken? Can any conclusions be drawn about the appropriate sample size?

10. A drug manufacturer wishes to control statistically the production of an antibiotic by taking samples periodically. It is known from production records that s=10mg. The maximum allowable error is two mg and the 99% confidence level is specified. How large a sample is necessary?

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Practice Problems

11. A firm wishes to estimate with an error of no more than .03 and a level of confidence of 98% the proportion of consumers that prefers its brand of detergent. Sales reports indicate that about 20% of all consumers prefer the firm’s brand. How large should the sample size be?

12. A study is made to estimate the average length of calls made to a call center. The mean length of a call is to be established within .30 minutes with a confidence level of 95%. Past studies show the standard deviation of call lengths to be approximately .9 minutes. How large should the sample size be?

13. The pharmacy wants to determine how many incoming shipments to sample based upon an error rate of 2%. They want the results to be accurate to within .1% with 99% confidence. How large should the sample be?

confidence and confidence intervals

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