Presentation on theme: "Estimating Population Values"— Presentation transcript:
1Estimating Population Values Chapter 7Estimating Population Values
2Chapter 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.
3Chapter 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.
4Point EstimatesA point estimate is a single number determined from a sample that is used to estimate the corresponding population parameter.
5Sampling ErrorSampling error refers to the difference between a value (a statistic) computed from a sample and the corresponding value (a parameter) computed from a population.
6Confidence IntervalsA 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.
9Confidence Interval - General Format - Point Estimate (Critical Value)(Standard Error)
10Confidence IntervalsThe 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.
11Confidence IntervalsThe confidence coefficient refers to the confidence level divided by 100% -- i.e., the decimal equivalent of a confidence level.
12Confidence Interval - General Format: known - Point Estimate z (Standard Error)
13Confidence Interval Estimates CONFIDENCE INTERVAL ESTIMATE FOR ( KNOWN)where:z = Critical value from standard normal table = Population standard deviationn = Sample size
14Example of a Confidence Interval Estimate for A random sample of 100 cans, from a population with = 0.20, produced a sample mean equal to A 95% confidence interval would be:ouncesounces
15Special 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.
16Margin of ErrorThe margin of error is the largest possible sampling error at the specified level of confidence.
17MARGIN OF ERROR (ESTIMATE FOR WITH KNOWN) where:e = Margin of errorz = Critical value= Standard error of the sampling distribution
18Example 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 A 95% confidence interval would be:ouncesouncesn=400n=100ouncesounces
19Student’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.
20Degrees of freedomDegrees 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.
21t-Values t-VALUE where: = Sample mean = Population mean s = Sample standard deviationn = Sample size
22Confidence Interval Estimates ( UNKNOWN)where:t = Critical value from t-distribution with n-1 degrees of freedom= Sample means = Sample standard deviationn = Sample size
23Confidence Interval Estimates CONFIDENCE INTERVAL-LARGE SAMPLE WITH UNKNOWNwhere:z =Value from the standard normal distribution= Sample means = Sample standard deviationn = Sample size
24Determining the Appropriate Sample Size SAMPLE SIZE REQUIREMENT - ESTIMATING WITH KNOWNwhere:z = Critical value for the specified confidence intervale = Desired margin of error = Population standard deviation
25Pilot SamplesA 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.
26Example 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.
27Estimating A Population Proportion SAMPLE PROPORTIONwhere:x = Number of occurrencesn = Sample size
28Estimating a Population Proportion STANDARD ERROR FOR pwhere: =Population proportionn = Sample size
29Confidence Interval Estimates for Proportions CONFIDENCE INTERVAL FOR where:p = Sample proportionn = Sample sizez = Critical value from the standard normal distribution
30Example 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:0.540.70
31Determining the Required Sample Size MARGIN OF ERROR FOR ESTIMATING where: = Population proportionz = Critical value from standard normal distributionn = Sample size
32Determining the Required Sample Size SAMPLE SIZE FOR ESTIMATING where: = Value used to represent the population proportione = Desired margin of errorz = Critical value from the standard normal table
33Key Terms Confidence Coefficient Confidence Interval Confidence Level Degrees of FreedomMargin of ErrorPilot SamplePoint EstimateSampling ErrorStudent’s t-distribution