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Interval Estimation Interval estimation of a population mean: Large Sample case Interval estimation of a population mean: Small sample case.

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Interval Estimation of a Population Mean: Large-Sample Case Sampling Error Probability Statements about the Sampling Error Interval Estimation: Assumed Known Interval Estimation: Estimated by s

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The absolute value of the difference between an The absolute value of the difference between an unbiased point estimate and the population unbiased point estimate and the population parameter it estimates is called the sampling error. parameter it estimates is called the sampling error. The absolute value of the difference between an The absolute value of the difference between an unbiased point estimate and the population unbiased point estimate and the population parameter it estimates is called the sampling error. parameter it estimates is called the sampling error. For the case of a sample mean estimating a For the case of a sample mean estimating a population mean, the sampling error is population mean, the sampling error is For the case of a sample mean estimating a For the case of a sample mean estimating a population mean, the sampling error is population mean, the sampling error is Sampling Error

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A probability statement about the sampling error is A probability statement about the sampling error is a precision statement. a precision statement. A probability statement about the sampling error is A probability statement about the sampling error is a precision statement. a precision statement. Knowledge of the sampling distribution of Knowledge of the sampling distribution of enables us to make probability statements about enables us to make probability statements about the sampling error even though the population the sampling error even though the population mean is not known. mean is not known. Knowledge of the sampling distribution of Knowledge of the sampling distribution of enables us to make probability statements about enables us to make probability statements about the sampling error even though the population the sampling error even though the population mean is not known. mean is not known. Probability Statements About the Sampling Error

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There is a 1 probability that the value of a sample mean will provide a sampling error of or less. Precision Statement /2 1 - of all values 1 - of all values Sampling distribution of Sampling distribution of

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/2 1 - of all values 1 - of all values Sampling distribution of Sampling distribution of [------------------------- -------------------------] interval does not include interval includes interval Interval Estimate of a Population Mean: Large-Sample Case ( n > 30)

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Assumed Known where: is the sample mean 1 - is the confidence coefficient 1 - is the confidence coefficient z /2 is the z value providing an area of z /2 is the z value providing an area of /2 in the upper tail of the standard /2 in the upper tail of the standard normal probability distribution normal probability distribution is the population standard deviation is the population standard deviation n is the sample size n is the sample size

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Interval Estimate of a Population Mean: Large-Sample Case (n > 30) Estimated by s In most applications the value of the population standard deviation is unknown. We simply use the value of the sample standard deviation, s, as the point estimate of the population standard deviation.

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Discount Sounds has 260 retail outlets throughout the United States. The firm is evaluating a potential location for a new outlet, based in part, on the mean annual income of the individuals in the marketing area of the new location. A sample of size n = 36 was taken. The sample mean income is $21,100 and the sample standard deviation is $4,500. The confidence coefficient to be used in the interval estimate is.95. Example: Discount Sounds S D

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95% of the sample means that can be observed are within + 1.96 of the population mean . Using s as an approximation of , the margin of error is: Precision Statement There is a.95 probability that the value of a There is a.95 probability that the value of a sample mean for Discount Sounds will provide sample mean for Discount Sounds will provide a sampling error of $1,470 or less. a sampling error of $1,470 or less. S D

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Interval estimate of is: Interval Estimate of Population Mean: Estimated by s S D We are 95% confident that the interval contains the population mean. $21,100 + $1,470 or $19,630 to $22,570

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Using Excel to Construct a Confidence Interval: Large-Sample Case n Formula Worksheet Note: Rows 15-37 are not shown. S D

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n Value Worksheet Note: Rows 15-37 are not shown. Using Excel to Construct a Confidence Interval: Large-Sample Case S D

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Interval Estimation of a Population Mean: Small-Sample Case (n < 30) Population is Not Normally Distributed The only option is to increase the sample size to n > 30 and use the large-sample interval-estimation procedures.

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Interval Estimation of a Population Mean: Small-Sample Case (n < 30) Population is Normally Distributed: Assumed Known The large-sample interval-estimation procedure can The large-sample interval-estimation procedure can be used.

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Population is Normally Distributed: Estimated by s Interval Estimation of a Population Mean: Small-Sample Case (n < 30) The appropriate interval estimate is based on a probability distribution known as the t distribution.

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The t distribution is a family of similar probability The t distribution is a family of similar probability distributions. distributions. The t distribution is a family of similar probability The t distribution is a family of similar probability distributions. distributions. t Distribution A specific t distribution depends on a parameter A specific t distribution depends on a parameter known as the degrees of freedom. known as the degrees of freedom. A specific t distribution depends on a parameter A specific t distribution depends on a parameter known as the degrees of freedom. known as the degrees of freedom. As the number of degrees of freedom increases, the As the number of degrees of freedom increases, the difference between the t distribution and the difference between the t distribution and the standard normal probability distribution becomes standard normal probability distribution becomes smaller and smaller. smaller and smaller. As the number of degrees of freedom increases, the As the number of degrees of freedom increases, the difference between the t distribution and the difference between the t distribution and the standard normal probability distribution becomes standard normal probability distribution becomes smaller and smaller. smaller and smaller. A t distribution with more degrees of freedom has A t distribution with more degrees of freedom has less dispersion. less dispersion. A t distribution with more degrees of freedom has A t distribution with more degrees of freedom has less dispersion. less dispersion.

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Degrees of Freedom Degrees of freedom (df) refers to the number of independent pieces of information that go into the computation of

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Example 17-19.6 22-14.6 34-2.6 5619.4 ?17.4 0 Note that Note that the 5 th value of x must be 54—given the values of x 1,...x 4. Thus 5-1 or 4 values of x are independent.

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t Distribution Standardnormaldistribution t distribution (20 degrees of freedom) t distribution (10 degrees of freedom) of freedom) 0 z, t

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/2 Area or Probability in the Upper Tail t Distribution 0 0 /2 t t t /2

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Interval Estimate Interval Estimation of a Population Mean: Small-Sample Case ( n < 30) and Estimated by s where: 1 - = the confidence coefficient t /2 = the t value providing an area of /2 t /2 = the t value providing an area of /2 in the upper tail of a t distribution in the upper tail of a t distribution with n - 1 degrees of freedom with n - 1 degrees of freedom s = the sample standard deviation s = the sample standard deviation

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Example: Apartment Rents Interval Estimation of a Population Mean: Small-Sample Case (n < 30) with Estimated by s A reporter for a student newspaper is writing an article on the cost of off-campus housing. A sample of 10 studio apartments within a half-mile of campus resulted in a sample mean of $550 per month and a sample standard deviation of $60.

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Example: Apartment Rents n Interval Estimation of a Population Mean: Small-Sample Case ( n < 30) with Estimated by s Small-Sample Case ( n < 30) with Estimated by s Let us provide a 95% confidence interval estimate of the mean rent per month for the population of studio apartments within a half-mile of campus. We will assume this population to be normally distributed.

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Interval Estimate Interval Estimation of a Population Mean: Small-Sample Case (n < 30) and Estimated by s where: 1 - = the confidence coefficient t /2 = the t value providing an area of /2 t /2 = the t value providing an area of /2 in the upper tail of a t distribution in the upper tail of a t distribution with n - 1 degrees of freedom with n - 1 degrees of freedom s = the sample standard deviation s = the sample standard deviation

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Example: Apartment Rents n Interval Estimation of a Population Mean: Small-Sample Case ( n < 30) with Estimated by s Small-Sample Case ( n < 30) with Estimated by s Let us provide a 95% confidence interval estimate of the mean rent per month for the population of studio apartments within a half-mile of campus. We will assume this population to be normally distributed.

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At 95% confidence, =.05, and /2 =.025. Interval Estimation of a Population Mean: Small-Sample Case (n < 30) and Estimated by s In the t distribution table we see that t.025 = 2.262. t.025 is based on n 1 = 10 1 = 9 degrees of freedom.

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Interval Estimation of a Population Mean: Small-Sample Case (n < 30) and Estimated by s We are 95% confident that the mean rent per We are 95% confident that the mean rent per month for the population of studio apartments within a half-mile of campus is between $507.08 and $592.92. n Interval Estimate = 550 + 42.92

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Using Excel to Construct a Confidence Interval: Estimated by s n Formula Worksheet

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n Value Worksheet Using Excel to Construct a Confidence Interval: Estimated by s

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Excel’s Descriptive Statistics tool can also be used to compute the margin of error when the t distribution is used for a confidence interval estimate of a population mean. Using Excel’s Descriptive Statistics Tool

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n Descriptive Statistics Dialog Box Using Excel’s Descriptive Statistics Tool

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n Value Worksheet

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Exercise 15, p. 334 The following data were collected for a sample from a normal population: 10, 8, 12, 15, 13, 11, 6, 5 a.What is the point estimate the population mean? b.What is the point estimate of the population standard deviation? c.What is the 95 percent confidence interval for the point estimate of the mean?

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Exercise 15, p. 334 (a) (b) (c)

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Summary of Interval Estimation Procedures for a Population Mean n > 30 ? s known ? Popul. approx.normal ? s known ? Use s to estimate s Use s to estimate s Increase n to > 30 Yes Yes Yes Yes No No No No

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