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QSO 510 Module Five 1

Module Four explored relationships between two quantitative variables using correlation analysis and developed prediction models using simple linear regression. Module Five examines point estimates and confidence interval estimates. A point estimate is a sample value, a statistic, used to estimate a population value, which is a parameter that is usually unknown. For example, the management of an amusement park wants an estimate of the mean amount spent by visitors to the park. It would not be feasible to contact each visitor, so 1,000 visitors are randomly selected as they leave the park and asked about their spending while visiting the park. The mean amount spent by the sample of 1,000 visitors serves as an estimate of the unknown average spending for the population of all visitors. While a point estimate is useful in describing a population, it is still a single value. A more informative estimate presents a range of values in which we expect the population parameter to occur. This range of values is called a confidence interval. A confidence interval is a range of values constructed from sample data so that the parameter occurs within that range with a specified probability. The specified probability is called the confidence level. A confidence interval consists of a statistic ± margin of error resulting in a lower limit and an upper limit. For example, we estimate that the mean annual income for recent MBA graduates in Houston, Texas, is $65,000. The margin of error is computed to be $4000. We can describe how confident we are that the population parameter is in the interval by making a probability statement. We might say that we are 90% confident that the mean annual income of recent MBA graduates in Houston, Texas, is $65,000 ±$4000, or between $61,000 and $69,000. Typical confidence levels are 99%, 95%, and 90%. The confidence interval that is most frequently used in practice is a confidence interval on a mean, of which there are two versions. If the population standard deviation is known, then a z confidence interval applies with this formula:

x bar ± z (σ/√n),

where x bar is a sample mean, z is a standard normal value, σ is a population standard deviation, and n is the size of the sample.

2 QSO 510 Module Five

For example, for a sample of 44 employees, the mean length of time that they are employed with a company is 10.455 years. From past data, the population standard deviation is known to be 7.7 years. A 90% confidence interval for the mean number of years employed is as follows:

10.455 ± 1.645(7.7/√44) = 10.455 ± 1.91 = (8.545, 12.365)

The 95% confidence interval for the mean number of years employed is 8.545 years to 12.365 years. If the population standard deviation is unknown, a t confidence interval is used with this formula:

x bar ± t (s/√n),

where t is a critical value from a standard t distribution and s is a sample standard deviation.

Confidence intervals have many real-world applications. A shoe manufacturer might use a 95% confidence interval of 9.5 inches to 9.7 inches to decide that a newly designed boot with a length of 10 inches would not meet a department store’s requirement and should be reworked. A production supervisor might use a 95% confidence interval of 0.7 cm to 1.0 cm to decide that a recently arrived piston of length 1.07 cm could not be used to repair a damaged oil pump on the shop floor.