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11.1 – Inference for Mean of a Population

11.1 – Inference for Mean of a Population. Conditions. Data are a SRS of size n from the population of interest Observations from the population have a normal distribution with mean µ and standard deviation σ . . Standard Error.

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11.1 – Inference for Mean of a Population

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1. 11.1 – Inference for Mean of a Population

2. Conditions • Data are a SRS of size n from the population of interest • Observations from the population have a normal distribution with mean µ and standard deviation σ.

3. Standard Error • It is an unrealistic assumption that we know the value of σ. • Because we don’t know σ, we estimate it by the sample standard deviation s. • We then estimate the standard deviation of x by s/√n : the standard error.

4. t-statistic • When we know σ, we base confidence intervals and tests for on the one-sample z-statistic: • When we do not know σ, we base confidence intervals and tests for on the one-sample t-statistic: • The one-sample t statistic has the t-distribution with n-1 degrees of freedom.

5. Degrees of Freedom • Sample standard deviation has n-1 degrees of freedom • As degrees of freedom, k, increases, the t(k) density curve approaches the N(0,1) more closely. • This is because s estimates σ more accurately when the sample size increases.

6. Example • A coffee machine dispenses coffee into paper cups. You’re supposed to get 10 ounces of coffee, but the amount varies slightly from cup to cup. Here are the amounts measured in a random sample of 20 cups. Is there evidence that the machine is shortchanging customers? 9.9 9.7 10.0 10.1 9.9 9.6 9.8 9.8 10.0 9.5 9.5 9.9 9.7 10.1 9.9 9.6 10.2 9.8 10.0 9.9

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