LESSON 18 CONFIDENCE INTERVAL ESTIMATION Outline Confidence interval

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LESSON 18: CONFIDENCE INTERVAL ESTIMATION Outline • Confidence interval: mean – Known σ –

LESSON 18: CONFIDENCE INTERVAL ESTIMATION Outline • Confidence interval: mean – Known σ – Selecting sample size – Unknown σ – Small population • Confidence interval: proportion • Confidence interval: variance 1

ESTIMATION • Point estimator: A point estimator draws inferences about a population by estimating

ESTIMATION • Point estimator: A point estimator draws inferences about a population by estimating the value of an unknown parameter using a single value or point. • Interval estimator: An interval estimator draws inferences about a population by estimating the value of an unknown parameter using an interval. 2

ESTIMATION • Example: A manager of a plant making cellular phones wants to estimate

ESTIMATION • Example: A manager of a plant making cellular phones wants to estimate the time to assemble a phone. A sample of 30 assemblies show a mean time of 400 seconds. The sample mean time of 400 seconds is a point estimate. An alternate estimate is a range e. g. , 390 to 410. Such a range is an interval estimate. The computation method of interval estimate is discussed in Chapter 10. 3

ESTIMATION • Interval estimates are reported with the end points e. g. , [390,

ESTIMATION • Interval estimates are reported with the end points e. g. , [390, 410] or, equivalently, with a central value and its difference from each end point e. g. , 400± 10 4

ESTIMATION • Precision of an interval estimate: The limits indicate the degree of precision.

ESTIMATION • Precision of an interval estimate: The limits indicate the degree of precision. A more precise estimate is the one with less spread between limits e. g. , [395, 405] or 400± 5 • Reliability of an interval estimate: The reliability of an interval estimate is the probability that it is correct. 5

ESTIMATION • Unbiased estimator: an unbiased estimator of a population parameter is an estimator

ESTIMATION • Unbiased estimator: an unbiased estimator of a population parameter is an estimator whose expected value is equal to that parameter. • In Chapter 2, the sample variance is defined as follows: • The use of n-1 in the denominator is necessary to get an unbiased estimator of variance. The use of n in the denominator produces a smaller value of variance. 6

ESTIMATION • Consistent estimators: An estimator is consistent if the precision and reliability improves

ESTIMATION • Consistent estimators: An estimator is consistent if the precision and reliability improves as the sample size is increased. The estimators are consistent. • Efficient estimators: An estimator is more efficient than another if for the sample size it will provide a greater sampling precision and reliability. 7

INTERVAL ESTIMATOR OF MEAN (KNOWN σ) • For some confidence level 1 - ,

INTERVAL ESTIMATOR OF MEAN (KNOWN σ) • For some confidence level 1 - , sample size n, sample mean, and the population standard deviation, the confidence interval estimator of mean, is as follows: • Recall that tail is /2 is that value of z for which area in the upper • Lower confidence limit (LCL) • Upper confidence limit (UCL) 8

CONFIDENCE INTERVAL 9

CONFIDENCE INTERVAL 9

AREAS FOR THE 82% CONFIDENCE INTERVAL 10

AREAS FOR THE 82% CONFIDENCE INTERVAL 10

AREAS AND z AND X VALUES FOR THE 82% CONFIDENCE INTERVAL 11

AREAS AND z AND X VALUES FOR THE 82% CONFIDENCE INTERVAL 11

INTERVAL ESTIMATOR OF MEAN (KNOWN σ) • Interpretation: – There is (1 - )

INTERVAL ESTIMATOR OF MEAN (KNOWN σ) • Interpretation: – There is (1 - ) probability that the sample mean will be equal to a value such that the interval (LCL, UCL) will include the population mean – If the same procedure is used to obtain a confidence interval estimate of the population mean for a sufficiently large number of k times, the interval (LCL, UCL) is expected to include the population mean (1 - )k times • Wrong interpretation: It’s wrong to interpret that there is (1 - ) probability that the population mean lies between LCL and UCL. Population mean is fixed, not uncertain / probabilistic. 12

INTERVAL ESTIMATOR OF MEAN (KNOWN σ) • Interpretation of the 95% confidence interval: –

INTERVAL ESTIMATOR OF MEAN (KNOWN σ) • Interpretation of the 95% confidence interval: – There is 0. 95 probability that the sample mean will be equal to a value such that the interval (LCL, UCL) will include the population mean – If the same procedure is used to obtain a confidence interval estimate of the population mean for a sufficiently large number of k times, the interval (LCL, UCL) is expected to include the population mean 0. 95 k times – • Wrong interpretation: It’s wrong to interpret that there is 0. 95 probability that the population mean lies between LCL and UCL. Population mean is fixed, not uncertain / probabilistic. 13

INTERVAL ESTIMATOR OF MEAN (KNOWN σ) Example 1: The following data represent a random

INTERVAL ESTIMATOR OF MEAN (KNOWN σ) Example 1: The following data represent a random sample of 10 observations from a normal population whose standard deviation is 2. Estimate the population mean with 90% confidence: 7, 3, 9, 11, 5, 4, 8, 3, 10, 9 14

SELECTING SAMPLE SIZE • A narrow confidence interval is more desirable. • For a

SELECTING SAMPLE SIZE • A narrow confidence interval is more desirable. • For a given a confidence level, a narrow confidence interval can be obtained by increasing the sample size. • Desired precision or maximum error: If the confidence interval has the form of then, d is the desired precision or the maximum error. • For a given confidence level (1 - ), desired precision d and the population standard deviation the sample size necessary to estimate population mean, is An approximation for : 15

SELECTING SAMPLE SIZE Example 2: Determine the sample size that is required to estimate

SELECTING SAMPLE SIZE Example 2: Determine the sample size that is required to estimate a population mean to within 0. 2 units with 90% confidence when the standard deviation is 1. 0. 16

INTERVAL ESTIMATOR OF MEAN (UNKNOWN σ) • If the population standard deviation σ is

INTERVAL ESTIMATOR OF MEAN (UNKNOWN σ) • If the population standard deviation σ is unknown, the normal distribution is not appropriate and the mean is estimated using Student t distribution. Recall that • For some confidence level 1 - , sample size n, sample mean, and the sample standard deviation, s the confidence interval estimator of mean, is as follows: • Where, is that value of t for which area in the upper tail 17 is /2 at degrees of freedom, d. f. = n-1.

SMALL POPULATION • For small, finite population, a correction factor is applied in computing.

SMALL POPULATION • For small, finite population, a correction factor is applied in computing. So, the confidence interval is computed as follows: 18

UNKNOWN σ AND SMALL POPULATION Example 3: An inspector wishes to estimate the mean

UNKNOWN σ AND SMALL POPULATION Example 3: An inspector wishes to estimate the mean weight of the contents in a shipment of 16 -ounce cans of corn. The shipment contains 500 cans. A sample of 25 cans is selected, and the contents of each are weighed. The sample mean and standard deviation were compute to be ounces and ounce. Construct a 90% confidence interval of the population mean. 19

INTERVAL ESTIMATOR OF PROPORTION • Confidence interval of the proportion for large population: •

INTERVAL ESTIMATOR OF PROPORTION • Confidence interval of the proportion for large population: • Confidence interval of the proportion for small population: • Required sample size for estimating the proportion: 20

INTERVAL ESTIMATOR OF PROPORTION Example 4: The controls in a brewery need adjustment whenever

INTERVAL ESTIMATOR OF PROPORTION Example 4: The controls in a brewery need adjustment whenever the proportion π of unfulfilled cans is 0. 01 or greater. There is no way of knowing the true proportion, however. Periodically, a sample of 100 cans is selected and the contents are measured. (a) For one sample, 3 under-filled can were found. Construct the resulting 95% confidence interval estimate of π. (b) What is probability of getting as many or more underfilled cans as in (a) when in fact π is only 0. 01. 21

INTERVAL ESTIMATOR OF VARIANCE • The chi-square distribution is asymmetric. As a result, two

INTERVAL ESTIMATOR OF VARIANCE • The chi-square distribution is asymmetric. As a result, two critical values are required to compute the confidence interval of the variance. • Confidence interval of the variance: 22

INTERVAL ESTIMATOR OF VARIANCE Example 5: The sample standard deviation for n = 25

INTERVAL ESTIMATOR OF VARIANCE Example 5: The sample standard deviation for n = 25 observations was computed to be s = 12. 2. Construct a 98% confidence interval estimate of the population standard deviation. 23

READING AND EXERCISES Lesson 18 Reading: Section 10 -1 to 10 -4, pp. 295

READING AND EXERCISES Lesson 18 Reading: Section 10 -1 to 10 -4, pp. 295 -319 Exercises: 10 -9, 10 -10, 10 -13, 10 -21, 10 -24, 10 -26, 10 -31, 10 -32 24