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t distribution with 20 degrees of freedom t distribution with 10 degrees of freedom 0 t distribution with degrees of freedom t Distribution The t-distribution is used when n is small and is unknown. Standard normal distribution

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0 t.0250 = ? t Distribution If n is large there is no difference between the standard normal and t distributions. t distribution with degrees of freedom.0250 Standard normal distribution

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t Distribution If n is large there is no difference between the standard normal and t distributions. P ( > 1.96) =.0250 t

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0 1.96 t Distribution t distribution with degrees of freedom.0250 If n is large there is no difference between the standard normal and t distributions. P ( > 1.96) =.0250 t Standard normal distribution

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Z.00.01.02.03.04.05.06.07.08.09 -2.2.0139.0136.0132.0129.0125.0122.0119.0116.0113.0110 -2.1.0179.0174.0170.0166.0162.0158.0154.0150.0146.0143 -2.0.0228.0222.0217.0212.0207.0202.0197.0192.0188.0183 -1.9.0287.0281.0274.0268.0262.0256.0250.0244.0239.0233 -1.8.0359.0351.0344.0336.0329.0322.0314.0307.0301.0294 -1.7.0446.0436.0427.0418.0409.0401.0392.0384.0375.0367 t Distribution If n is large there is no difference between the standard normal and t distributions. P ( > 1.96) =.0250 t z

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t Distribution If n is large there is no difference between the standard normal and t distributions.

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0 1.96 t Distribution If n is large there is no difference between the standard normal and t distributions. t distribution with degrees of freedom.0250 Standard normal distribution

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0 t Distribution If n is large there is no difference between the standard normal and t distributions. t.0250 = ?.0250 with df = 20

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t Distribution P ( > 2.086) =.0250 t If n is large there is no difference between the standard normal and t distributions. with df = 20

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0 t Distribution If n is large there is no difference between the standard normal and t distributions. 2.086.0250 with df = 20

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0 t Distribution with df = 10 If n is large there is no difference between the standard normal and t distributions..0250 t.0250 = ?

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t Distribution If n is large there is no difference between the standard normal and t distributions. with df = 10

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0 t Distribution with df = 10 If n is large there is no difference between the standard normal and t distributions..0250 2.228

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Interval Estimate of a Population Mean is the known population standard deviation z /2 is the z value providing an area of /2 in the upper tail of the standard normal distribution OR n > 2if the data is normally distributed n > 30if the data is roughly symmetric n > 50if the data is heavily skewed and The 1 confidence interval for unknown

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Interval Estimate of a Population Proportion z /2 is the z value providing an area of /2 in the upper tail of the standard normal distribution and n > 5/p n > 5/(1 – p) and The 1 confidence interval for unknown p

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Interval Estimate of a Population Mean The 1 confidence interval for unknown s estimates because it is unknown t /2 is the t value providing an area of /2 in the upper tail of the t-distribution OR n > 2if the data is normally distributed n > 15if the data is roughly symmetric n > 30if the data is slightly skewed n > 50if the data is heavily skewed and OR

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Example: 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 of 36 was taken; the sample mean income is $31,100. The population is not believed to be highly skewed. The population standard deviation is estimated to be $4,500, and the confidence coefficient to be used in the interval estimate is.95. n x Interval Estimate of a Population Mean

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The margin of error is: Thus, at 95% confidence, the margin of error is $1,470. Interval estimate of is: We are 95% confident that the interval contains. $31,100 + $1,470 $29,630 to $32,570 Interval Estimate of a Population Mean Example: Discount Sounds

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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. Example: Political Science, Inc. In a current election campaign, PSI has just found that 220 registered voters, out of 500 contacted, favor McSame. PSI wants to develop a 95% confidence interval estimate for the proportion of the population of registered voters that favor McSame. n Interval Estimate of a Population Proportion

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where: n = 500, PSI is 95% confident that the proportion of all voters that favor McSame is between.3965 and.4835. =.44 +.0435 = 220/500 =.44,z = 1.96 Interval Estimate of a Population Proportion Example: Political Science, Inc.

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A reporter for a student newspaper is writing an article on the cost of off-campus housing. A sample of 16 efficiency apartments within a half-mile of campus resulted in a sample mean of $650 per month and a sample standard deviation of $55. Example: Apartment Rents Compute a 95% confidence interval estimate of the mean rent per month for the population of efficiency apartments within a half-mile of campus. We will assume this population to be normally distributed. s x n Interval Estimate of a Population Mean

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At 95% confidence, t.025 is based on df = n 1 = =.05, /2 =.025. 16 1 = 15 Interval Estimate of a Population Mean

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We are 95% confident that the mean rent per month for the population of efficiency apartments within a half-mile of campus is between $620.70 and $679.30. Interval Estimate Est. Margin of Error Point Estimate of the population mean Interval Estimate of the population mean Interval Estimate of a Population Mean

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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 probability that the margin of error is $500 or less. How large should the sample be? E n?n? Recall that = 4,500. Sufficient Sample Size for the Interval Estimate of a Population Mean At 95% confidence, z.025 = 1.96.

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Suppose that PSI would like a.99 probability that the sample proportion is within.03 of the population proportion. How large should the sample be to meet the required precision? Recall that in the previous example, a sample of 500 similar units yielded a sample proportion of.44. E p*p* Example: Political Science, Inc. Sufficient Sample Size for the Interval Estimate of a Population Proportion At 99% confidence, =.01, /2 =.005, z.005 = 2.575

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.025.950 = 10 n = 64 = 0.05 z.025 = 1.96 Repeated Sampling

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.025.950 = 10 n = 64 = 0.05 z.025 = 1.96 Repeated Sampling

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.025.950 = 10 n = 64 = 0.05 z.025 = 1.96 Repeated Sampling Example: Political Science, Inc.

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