Lecture7.chapt7

# Lecture7.chapt7 - Lecture 7 Sections 7.1& 7.2 Inference...

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Unformatted text preview: Lecture 7, Sections 7.1 & 7.2 Inference for the Mean of a Population Previously we made the assumption that we know the population standard deviation, σ . We then developed a confidence interval and used tests for significance to gather evidence for/against an hypothesis, all with a known σ . In normal practice, σ is unknown. In this section, we must estimate σ from the data though we are primarily interested in the population mean, µ . Confidence Interval for a Mean First, Assumptions for Inference about a mean: • Our data are a simple random sample (SRS) of size n from the population. • Observations from the population have a normal distribution with mean µ and standard deviation σ . If population distribution is not normal, it is enough that the distribution is unimodal and symmetric and that the sample size be large (n>15). Both µ and σ are unknown parameters. Because we do not know σ we make two changes in our procedure. 1. The standard error, x SE , is used in place of n σ . Standard Error: When the standard deviation of a statistic is estimated from the data, the result is called the standard error of the statistic. The standard error of the sample mean x is x s SE n = Where s is the sample standard deviation, n is the sample size. 2. We calculate a different test statistic and use a different distribution to calculate our p-value. The t-distributions: Lecture 7, Section 7.1 & 7.2 Page 1 • The t-distribution is used when we do not know σ . The t-distributions have density curves similar in shape to the standard normal curve, but with more spread. • The t-distributions have more probability in the tails and less in the center than does the standard normal. This is because substituting the estimate s for the fixed parameter σ introduces more variation into the statistic. • As the sample size increases, the t-density curve approaches the N(0,1) curve. (Note: This is because s estimates σ more accurately as the sample size increases). The t Distributions Suppose that an SRS of size n is drawn from a ( , ) N μ σ population. Then the one- sample t statistic / x t s n μ- = has the t distribution with n-1 degrees of freedom . The One-Sample t Confidence Interval Suppose that an SRS of size n is drawn from a population having unknown mean µ . A level C confidence interval for µ is * s x t n ± where t* is the value for the t ( n-1) density curve with area C between – t* and t* . This interval is exact when the population distribution is normal and is approximately correct for large n in other cases. Examples: Lecture 7, Section 7.1 & 7.2 Page 2 1. Suppose X, Bob’s golf scores, are approximately normal distribution with unknown mean and standard deviation. A SRS of n = 16 scores is selected and a sample mean of x = 77 and a sample standard deviation of s = 3 is calculated. Calculate a 90% confidence interval for μ ....
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Lecture7.chapt7 - Lecture 7 Sections 7.1& 7.2 Inference...

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