LESSON 18: ESTIMATION

Outline

• Confidence interval: – Known s – Selecting size – Unknown s – Small population • Confidence interval: proportion • Confidence interval:

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ESTIMATION

• Point : 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.

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1 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 e.g., 390 to 410. Such a range is an interval estimate. The computation method of interval estimate is discussed in Chapter 10.

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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

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2 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.

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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: n 2 å(X i - X ) s2 = i=1 n -1 • 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.

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3 ESTIMATION

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

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INTERVAL ESTIMATOR OF MEAN (KNOWN s)

• For some confidence level 1-a, sample size n, sample mean, X and the population , s the confidence interval estimator of mean, m is as follows: s é s s ù X ± za /2 also written as ê X - za /2 , X + za /2 ú n ë n n û

• Recall that z a / 2 is that value of z for which area in the upper tail is a/2

s • Lower confidence limit (LCL) X - za /2 n s • Upper confidence limit (UCL) X + za /2 n 8

4 Area=1-a CONFIDENCE s

) s = INTERVAL X x

( n f

Area Area =0.5-a/2 =0.5-a/2

Area=a/2 Area=a/2 -za/2 z=0 za/2

s s X - za /2 X X + za /2 n n 9

AREAS FOR Area=0.82 THE 82% s

) s = CONFIDENCE X x

( n INTERVAL f

Area Area =0.41 =0.41

Area=0.09 Area=0.09 -z0.09 z=0 z0.09

s s X - z0.09 X X + z0.09 n n 10

5 AREAS AND z Area=0.82 AND X s

VALUES FOR ) s X = x

( n THE 82% f

CONFIDENCE Area Area INTERVAL =0.41 =0.41

Area=0.09 Area=0.09 -1.34 z=0 1.34

s s X -1.34 X X +1.34 n n 11

INTERVAL ESTIMATOR OF MEAN (KNOWN s)

• Interpretation: – There is (1-a) 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-a)k times • Wrong interpretation: It’s wrong to interpret that there is (1-a) probability that the population mean lies between LCL and UCL. Population mean is fixed, not uncertain / probabilistic. 12

6 INTERVAL ESTIMATOR OF MEAN (KNOWN s)

• 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 0.95k 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 s)

Example 1: The following 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

s s x =

) n x ( f

x 14

7 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 X ± d then, d is the desired precision or the maximum error. • For a given confidence level (1-a), desired precision d and the population standard deviation s the sample size necessary to estimate population mean, m is 2 æ z s ö Q -Q n = ç a /2 ÷ An approximation for s: s = .999 .001 è d ø 6 15

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.

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8 INTERVAL ESTIMATOR OF MEAN (UNKNOWN s)

• If the population standard deviation s is unknown, the is not appropriate and the mean is estimated using Student t distribution. Recall that X - m t = s / n • For some confidence level 1-a, sample size n, sample mean, X and the sample standard deviation, s the confidence interval estimator of mean, m is as follows: s é s s ù X ±ta /2 also written as ê X -ta /2 , X +ta /2 ú n ë n n û

• Where, t a / 2 is that value of t for which area in the upper tail is a/2 at degrees of freedom, d.f. = n-1. 17

SMALL POPULATION

• For small, finite population, a correction factor is applied in

computing s X . So, the confidence interval is computed as follows: s N - n m = X ± z (s known ) a /2 n N -1

s N - n m = X ± ta /2 (s unknown ) n N -1

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9 UNKNOWN s 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 X = 16 . 1 ounces and s = 0 . 25 ounce. Construct a 90% confidence interval of the population mean.

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INTERVAL ESTIMATOR OF PROPORTION

• Confidence interval of the proportion for large population: P(1- P) p = P ± z a /2 n • Confidence interval of the proportion for small population: P(1- P) N - n p = P ± z a /2 n N -1 • Required sample size for estimating the proportion: z 2 p(1-p ) n = a /2 d 2

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10 INTERVAL ESTIMATOR OF PROPORTION

Example 4: The controls in a brewery need adjustment whenever the proportion p 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 p. (b) What is probability of getting as many or more under- filled cans as in (a) when in fact p is only 0.01.

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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: 2 2 (n -1)s 2 (n -1)s 2 £s £ 2 ca /2 c1-a /2

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11 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.

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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

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