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Unformatted text preview: Chapter 8 Interval Estimation
s s s Interval Estimation of a Population Mean • σ Known Case • Large Sample (n ≥ 30) • Small Sample (n < 30) • σ Unknown Case • Large Sample (n ≥ 30) • Small Sample (n < 30) x Determining the Sample Size µ Interval Estimation of a Population Proportion [ x ]
[ x ] [ x ] Slide 1 Summary of Confidence Interval Estimation Procedures for a Population Mean Slide 2 Margin of Error and the Interval Estimate A point estimator cannot be expected to provide the exact value of the population parameter. An interval estimate can be computed by adding and subtracting a margin of error to the point estimate. Point Estimate + Margin of Error Motivation for Interval Estimation: The purpose of an interval estimate is to provide information about how close the point estimate is to the value of the parameter (which is unknown). Slide 3 Margin of Error and the Interval Estimate The general form of an interval estimate of a population mean is x ± Margin of Error From last chapter, we know how to compute: From P(−k ≤ X − µ ≤ k ) Slide 4 There is a 1 − α probability that the value of a zα / 2 σ x sample mean will provide a margin of error of or less. Sampling distribution of x σ Known Case : Interval Estimate of a μ α/2 1 α of all x values α /2
x zα / 2 σ x µ zα / 2 σ x Slide 5 σ Known Case : Interval Estimate of a μ
Sampling distribution of x α/2
interval does not include µ 1 α of all x values α/2 zα / 2 σ x µ x
zα / 2 σ x [[ x ] interval includes µ [ x x ] ] Slide 6 σ
s Known : Interval Estimate of a μ Known : Interval Estimate of a Invoke Central Li mit Theorem
x ± zα / 2 LargeSample (n ≥ 30) x where: is the sample mean 1 α is the confidence coefficient zα/2 is the z value providing an area of α/2 in the upper tail of the standard normal probability distribution σ is the population standard deviation n is the sample size Slide 7 σ n σ
s Known : Interval Estimate of a μ Known : Interval Estimate of a Assume Normal Population
x ± zα / 2 SmallSample (n < 30) x where: is the sample mean 1 α is the confidence coefficient zα/2 is the z value providing an area of α/2 in the upper tail of the standard normal probability distribution σ is the population standard deviation n is the sample size NOTE: If the population is not normal – then you will have to increase the sample size to n ≥ 30. Slide 8 σ n σ
s Unknown : Interval Estimate of a μ Unknown : Interval Estimate of a Assume Normal Population LargeSample (n ≥ 30) In most applications the value of the population standard deviation is unknown. In this regard, we could either one of the expressions usingthe value of the sample standard deviation, s, as the point estimate of the population standard deviation. x ± zα / 2 s n OR x ± tα /2 s n NOTE: You can use either Z or t distribution NOTE: Slide 9 σ
s Unknown : Interval Estimate of a μ Unknown : Interval Estimate of a Assume Normal Population SmallSample (n < 30) • The appropriate interval estimate is based on a probability distribution known as the t distribution. • • X −µ In this case, follows t distribution. s/ n This also holds for n > 30 as shown in previous slide. Slide 10 σ
s Unknown : Interval Estimate of a μ Unknown : Interval Estimate of a Assume Normal Population
x ± tα / 2 s n Interval Estimate of µ where 1 α = the confidence coefficient tα/2 = the t value providing an area of α /2 in the upper tail of a t distribution with n 1 degrees of freedom s = the sample standard deviation Slide 11 t Distribution
The t distribution is a family of similar probability distributions. s A specific t distribution depends on a parameter known as the degrees of freedom. s A t distribution with more degrees of freedom has less dispersion. s As the number of degrees of freedom increases, the difference between the t distribution and the standard normal probability distribution becomes smaller and smaller. s The standard normal z values can be found in the ∞ infinite degrees ( ) row of the t distribution table. s The mean of the t distribution is always zero.
s Slide 12 t Distribution
t distribution (20 degrees of freedom) t distribution (10 degrees of freedom) z, t Standard normal distribution 0 Slide 13 t Distribution
D egrees of Freedom . 50 60 80 100 .20 . .849 .848 .846 .845 .842 .10 . 1.299 1.296 1.292 1.290 1.282 Area in Upper Tail .05 . 1.676 1.671 1.664 1.660 1.645 .025 . 2.009 2.000 1.990 1.984 1.960 .01 . 2.403 2.390 2.374 2.364 2.326 .005 . 2.678 2.660 2.639 2.626 2.576 ∞ Standard normal z values Slide 14 Interval Estimation of a Population Mean
s Example 1: Discount Sounds 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 $31,100 and the sample standard deviation is $4,500. The confidence coefficient to be used in the interval estimate is .95. D S Slide 15 Interval Estimate of Population Mean: Example 1: Discount Sounds
a. Calculate the margin of error. D
S b. Find the 95% confidence interval estimate of µ . We are 95% confident that the above interval contains the population mean. Slide 16 Interval Estimation of a Population Mean
s Example 2: Apartment Rents A reporter for a student newspaper is writing an article on the cost of offcampus housing. A sample of 16 efficiency apartments within a halfmile of campus resulted in a sample mean of $650 per month and a sample standard deviation of $55. Slide 17 Interval Estimation of a Population Mean
s Example 2: Apartment Rents Let us provide a 95% confidence interval estimate of the mean rent per month for the population of efficiency apartments within a halfmile of campus. We will assume this population to be normally distributed. Slide 18 Interval Estimation of a Population Mean
Q. Compute the 95% confidence interval for the population mean and interpret the result.
s We are 95% confident that the mean rent per month for the population of efficiency apartments within a halfmile of campus is between Slide 19 Interval Estimation of a Population Mean
At 95% confidence, α = .05, and α/2 = .025. t.025 is based on n − 1 = 16 − 1 = 15 degrees of freedom. In the t distribution table we see that t.025 = 2.131.
D e gre e s of Fre e dom 15 16 17 18 19 . .20 .866 .865 .863 .862 .861 . .100 1.341 1.337 1.333 1.330 1.328 . Are a in Uppe r Tail .050 1.753 1.746 1.740 1.734 1.729 . .025 2.131 2.120 2.110 2.101 2.093 . .010 2.602 2.583 2.567 2.520 2.539 . .005 2.947 2.921 2.898 2.878 2.861 . Slide 20 Summary of Confidence Interval Estimation Procedures for a Population Mean Slide 21 Sample Size for an Interval Estimate of a Population Mean Let E = the desired margin of error.
s Margin of Error E = zα / 2
s σ n Necessary Sample Size ( zα / 2 ) 2 σ 2 n= E2 Slide 22 Sample Size for an Interval Estimate of a Population Mean D
S Recall that Discount Sounds is evaluating a potential location for a new retail outlet, based in part, on the mean annual income of the individuals in the marketing area of the new location. Suppose that Discount Sounds’ management team wants an estimate of the population mean such that there is a .95 confidence that the sampling error is $500 or less. Q. How large a sample size is needed to meet the required precision? Slide 23 Interval Estimation of a Population Proportion The general form of an interval estimate of a population proportion is p ± Margin of Error p The sampling distribution of plays a key role in computing the margin of error for this interval estimate. p The sampling distribution of can be approximated by a normal distribution whenever np > 5 and n(1 – p) > 5. Slide 24 Interval Estimation of a Population Proportion
s Normal Approximation of Sampling Distribution of Sampling distribution of p p σp = p(1 − p ) n α /2 1 α of all p values p α/2 p zα / 2 σ p zα / 2 σ p Slide 25 Interval Estimation of a Population Proportion
s 100(1α)% Confidence Interval Estimate of p p ± zα / 2 p (1 − p ) n where: 1 α is the confidence coefficient zα/2 is the z value providing an area of α /2 in the upper tail of the standard normal probability distribution p is the sample proportion Slide 26 Interval Estimation of a Population Proportion
s Example: Political Science, Inc. Political Science, Inc. (PSI) specializes in voter polls and surveys designed to keep political office seekers informed of their position in a race. Using telephone surveys, PSI interviewers ask registered voters who they would vote for if the election were held that day. Slide 27 Interval Estimation of a Population Proportion
s Example: Political Science, Inc. In a current election campaign, PSI has just found that 220 registered voters, out of 500 contacted, favor a particular candidate. PSI wants to develop a 95% confidence interval estimate for the proportion of the population of registered voters that favor the candidate. Slide 28 Interval Estimation of a Population Proportion
Q. Compute the 95% confidence interval for the population proportion and interpret the result. PSI is 95% confident that the proportion of all voters that favor the candidate is between Slide 29 Sample Size for an Interval Estimate of a Population Proportion
s Margin of Error E = zα / 2 p (1 − p ) n Solving for the necessary sample size, we get ( zα / 2 ) 2 p (1 − p ) n= E2 p However, will not be known until after we have selected the sample. Slide 30 Sample Size for an Interval Estimate of a Population Proportion
Suppose that PSI would like a .99 probability that the sample proportion is within + .03 of the population proportion. Q. How large a sample size is needed to meet the required precision? (A previous sample of similar units yielded .44 for the sample proportion.) Slide 31 Sample Size for an Interval Estimate of a Population Proportion
Note: We used .44 as the best estimate of p in the preceding expression. If no information is available about p, then .5 is often assumed because it provides the highest possible sample size. If we had used p = .5, the recommended n would have been 1849. Slide 32 ...
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This note was uploaded on 12/04/2009 for the course MGMT 305 taught by Professor Priya during the Summer '08 term at Purdue UniversityWest Lafayette.
 Summer '08
 Priya

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