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**t Distribution t distribution with ∞ degrees of freedom**

The t-distribution is used when n is small and s is unknown. t distribution with ∞ degrees of freedom Standard normal distribution t distribution with 20 degrees of freedom The t distribution is a family of similar probability distributions that is no different from the standard normal distribution when the sample size is large. A specific t distribution depends on a parameter known as the degrees of freedom. Degrees of freedom refer to the number of independent pieces of information that go into the computation of s. t distribution with 10 degrees of freedom

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**t distribution with ∞ degrees of freedom**

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

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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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**t distribution with ∞ degrees of freedom**

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

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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 z 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

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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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**t distribution with ∞ degrees of freedom**

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

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**t Distribution with df = 20 t.0250 = ?**

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

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**t Distribution with df = 20**

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

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**t Distribution with df = 20 2.086**

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

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**t Distribution with df = 10 t.0250 = ?**

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

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**t Distribution with df = 10**

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

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**t Distribution with df = 10 2.228**

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

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**Interval Estimate of a Population Mean**

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

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**Interval Estimate of a Population Proportion**

The 1 - confidence interval for unknown p 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

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**Interval Estimate of a Population Mean**

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

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**Interval Estimate of a Population**

Mean 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. n x 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. s 1 - a

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**Interval Estimate of a Population**

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

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**Interval Estimate of a Population**

Proportion 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. 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 1 - a

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**Interval Estimate of a Population**

Proportion Example: Political Science, Inc. where: n = 500, = 220/500 = .44, z.025 = 1.96 = PSI is 95% confident that the proportion of all voters that favor McSame is between and

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**Interval Estimate of a Population**

Mean Example: Apartment Rents 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. n x s 1 - a 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.

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**Interval Estimate of a Population**

Mean At 95% confidence, = .05, /2 = .025. t.025 is based on df = n - 1 = = 15

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**Interval Estimate of a Population Mean**

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

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**Sufficient Sample Size for the Interval Estimate of a Population Mean**

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? 1-a n? E At 95% confidence, z.025 = 1.96. Recall that = 4,500.

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**Sufficient Sample Size for the Interval Estimate of a Population Proportion**

Example: Political Science, Inc. 1-a 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* At 99% confidence, a = .01, a/2 = .005, z.005 = 2.575 Using the sample proportion from a previous sample of the same or similar units Select a preliminary sample to compute a “preliminary” sample proportion. If no information is available about p, then .5 is often assumed because it provides the highest possible sample size.

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Repeated Sampling s = 10 n = 64 a = 0.05 z.025 = 1.96 .025 .950

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Repeated Sampling s = 10 n = 64 a = 0.05 z.025 = 1.96 .025 .950

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**Repeated Sampling Example: Political Science, Inc. s = 10 n = 64**

z.025 = 1.96 .025 .950

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