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

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.

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.

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

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.

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.

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 same sample size it will provide a greater sampling precision and reliability.

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 is that value of z for which area in the upper tail is /2 Lower confidence limit (LCL) Upper confidence limit (UCL)

CONFIDENCE INTERVAL

AREAS FOR THE 82% CONFIDENCE INTERVAL

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

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.

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 0.95k 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.

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

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 :

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.

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 is /2 at degrees of freedom, d.f. = n-1.

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

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.

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:

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. For one sample, 3 under-filled can were found. Construct the resulting 95% confidence interval estimate of π. What is probability of getting as many or more under-filled cans as in (a) when in fact π is only 0.01.

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:

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.

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