This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 pvalue. The tdistributions: Lecture 7, Section 7.1 & 7.2 Page 1 • The tdistribution is used when we do not know σ . The tdistributions have density curves similar in shape to the standard normal curve, but with more spread. • The tdistributions 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 tdensity 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 n1 degrees of freedom . The OneSample 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 ( n1) 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 ....
View
Full Document
 Spring '08
 Staff
 Statistics, Normal Distribution, Standard Deviation, Statistical hypothesis testing

Click to edit the document details