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**Estimating Population Values**

Chapter 7 Estimating Population Values

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**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.

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**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.

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

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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.

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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.

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**Lower Confidence Limit Upper Confidence Limit**

Confidence Intervals Lower Confidence Limit Upper Confidence Limit Point Estimate

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**95% Confidence Intervals (Figure 7-3)**

0.95 z.025= -1.96 z.025= 1.96

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**Confidence Interval - General Format -**

Point Estimate (Critical Value)(Standard Error)

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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.

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

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**Confidence Interval - General Format: known -**

Point Estimate z (Standard Error)

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**Confidence Interval Estimates**

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

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**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 A 95% confidence interval would be: ounces ounces

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**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.

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

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**MARGIN OF ERROR (ESTIMATE FOR WITH KNOWN)**

where: e = Margin of error z = Critical value = Standard error of the sampling distribution

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**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 A 95% confidence interval would be: ounces ounces n=400 n=100 ounces ounces

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**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.

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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.

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**t-Values t-VALUE where: = Sample mean = Population mean**

s = Sample standard deviation n = Sample size

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**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

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**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

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**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

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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.

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**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.

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**Estimating A Population Proportion**

SAMPLE PROPORTION where: x = Number of occurrences n = Sample size

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**Estimating a Population Proportion**

STANDARD ERROR FOR p where: =Population proportion n = Sample size

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**Confidence Interval Estimates for Proportions**

CONFIDENCE INTERVAL FOR where: p = Sample proportion n = Sample size z = Critical value from the standard normal distribution

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**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

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**Determining the Required Sample Size**

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

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**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

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**Key Terms Confidence Coefficient Confidence Interval Confidence Level**

Degrees of Freedom Margin of Error Pilot Sample Point Estimate Sampling Error Student’s t-distribution

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