Estimating Population Values Chapter 7 Estimating Population Values

Chapter 7 - Chapter Outcomes After studying the material in this chapter, you should be able to: Distinguish between a point estimate and a confidence interval estimate. Construct and interpret a confidence interval estimate for a single population mean using both the z and t distributions.

Chapter 7 - Chapter Outcomes (continued) After studying the material in this chapter, you should be able to: Determine the required sample size for an estimation application involving a single population mean. Establish and interpret a confidence interval estimate for a single population proportion.

Point Estimates A point estimate is a single number determined from a sample that is used to estimate the corresponding population parameter.

Sampling Error Sampling error refers to the difference between a value (a statistic) computed from a sample and the corresponding value (a parameter) computed from a population.

Confidence Intervals A confidence interval refers to an interval developed from randomly sample values such that if all possible intervals of a given width were constructed, a percentage of these intervals, known as the confidence level, would include the true population parameter.

Lower Confidence Limit Upper Confidence Limit Confidence Intervals Lower Confidence Limit Upper Confidence Limit Point Estimate

95% Confidence Intervals (Figure 7-3) 0.95 z.025= -1.96 z.025= 1.96

Confidence Interval - General Format - Point Estimate  (Critical Value)(Standard Error)

Confidence Intervals The confidence level refers to a percentage greater than 50 and less than 100 that corresponds to the percentage of all possible confidence intervals, based on a given size sample, that will contain the true population value.

Confidence Intervals The confidence coefficient refers to the confidence level divided by 100% -- i.e., the decimal equivalent of a confidence level.

Confidence Interval - General Format:  known - Point Estimate  z (Standard Error)

Confidence Interval Estimates CONFIDENCE INTERVAL ESTIMATE FOR  ( KNOWN) where: z = Critical value from standard normal table  = Population standard deviation n = Sample size

Example of a Confidence Interval Estimate for  A random sample of 100 cans, from a population with  = 0.20, produced a sample mean equal to 12.09. A 95% confidence interval would be: 12.051 ounces 12.129 ounces

Special Message about Interpreting Confidence Intervals Once a confidence interval has been constructed, it will either contain the population mean or it will not. For a 95% confidence interval, if you were to produce all the possible confidence intervals using each possible sample mean from the population, 95% of these intervals would contain the population mean.

Margin of Error The margin of error is the largest possible sampling error at the specified level of confidence.

MARGIN OF ERROR (ESTIMATE FOR  WITH  KNOWN) where: e = Margin of error z = Critical value = Standard error of the sampling distribution

Example of Impact of Sample Size on Confidence Intervals If instead of random sample of 100 cans, suppose a random sample of 400 cans, from a population with  = 0.20, produced a sample mean equal to 12.09. A 95% confidence interval would be: 12.0704 ounces 12.1096 ounces n=400 n=100 12.051 ounces 12.129 ounces

Student’s t-Distribution The t-distribution is a family of distributions that is bell-shaped and symmetric like the standard normal distribution but with greater area in the tails. Each distribution in the t-family is defined by its degrees of freedom. As the degrees of freedom increase, the t-distribution approaches the standard normal distribution.

Degrees of freedom Degrees of freedom refers to the number of independent data values available to estimate the population’s standard deviation. If k parameters must be estimated before the population’s standard deviation can be calculated from a sample of size n, the degrees of freedom are equal to n - k.

t-Values t-VALUE where: = Sample mean = Population mean s = Sample standard deviation n = Sample size

Confidence Interval Estimates ( UNKNOWN) where: t = Critical value from t-distribution with n-1 degrees of freedom = Sample mean s = Sample standard deviation n = Sample size

Confidence Interval Estimates CONFIDENCE INTERVAL-LARGE SAMPLE WITH  UNKNOWN where: z =Value from the standard normal distribution = Sample mean s = Sample standard deviation n = Sample size

Determining the Appropriate Sample Size SAMPLE SIZE REQUIREMENT - ESTIMATING  WITH  KNOWN where: z = Critical value for the specified confidence interval e = Desired margin of error  = Population standard deviation

Pilot Samples A pilot sample is a random sample taken from the population of interest of a size smaller than the anticipated sample size that is used to provide and estimate for the population standard deviation.

Example of Determining Required Sample Size (Example 7-7) The manager of the Georgia Timber Mill wishes to construct a 90% confidence interval with a margin of error of 0.50 inches in estimating the mean diameter of logs. A pilot sample of 100 logs yields a sample standard deviation of 4.8 inches. Note, the manager needs only 150 more logs since the 100 in the pilot sample can be used.

Estimating A Population Proportion SAMPLE PROPORTION where: x = Number of occurrences n = Sample size

Estimating a Population Proportion STANDARD ERROR FOR p where:  =Population proportion n = Sample size

Confidence Interval Estimates for Proportions CONFIDENCE INTERVAL FOR  where: p = Sample proportion n = Sample size z = Critical value from the standard normal distribution

Example of Confidence Interval for Proportion (Example 7-8) 62 out of a sample of 100 individuals who were surveyed by Quick-Lube returned within one month to have their oil changed. To find a 90% confidence interval for the true proportion of customers who actually returned: 0.54 0.70

Determining the Required Sample Size MARGIN OF ERROR FOR ESTIMATING  where:  = Population proportion z = Critical value from standard normal distribution n = Sample size

Determining the Required Sample Size SAMPLE SIZE FOR ESTIMATING  where:  = Value used to represent the population proportion e = Desired margin of error z = Critical value from the standard normal table

Key Terms Confidence Coefficient Confidence Interval Confidence Level Degrees of Freedom Margin of Error Pilot Sample Point Estimate Sampling Error Student’s t-distribution