501_Lecture_07

501_Lecture_07 - Section 7.1 Inference for the mean of a...

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Unformatted text preview: Section 7.1 Inference for the mean of a population Change: Population standard deviation (σ) is now unknown The t distribution One-sample t confidence interval One-sample t test Matched pairs t procedures Robustness of t procedures The t distribution: The goal is to estimate or test for an unknown µ in the situation when σ is also unknown. Solution: estimate σ by s and use it intelligently in the formulas. Challenge: the distribution of the test statistic will change and will no longer be the standard normal distribution. Sampling – Normal Population, Unknown Standard Deviation Suppose an SRS X 1 , …, X n is selected from a normally distributed population with mean μ and standard deviation σ. Assume that μ and σ are both unknown. We know that When σ is unknown, we estimate its value with the sample standard deviation s . ( 29 n N X σ μ , ~ Sampling – Normal Population, Unknown Standard Deviation The standard deviation of can be estimated by This quantity is called the standard error of the sample mean. The test statistic (appropriately standardized sample mean) will no longer be normally distributed when we use the standard error. The test statistic will have a new distribution, called the t (or Student’s t ) distribution. X n s SE X = The t-test Statistic and Distribution Suppose that an SRS of size n is drawn from an N ( μ , σ) population. Then the one-sample t- statistic has the t distribution with n – 1 degrees of freedom. There is a different t distribution for each sample size. The degrees of freedom for the t-statistic “come” from the sample standard deviation s . The density curve of a t distribution with k degrees of freedom is symmetric about 0 and bell-shaped. x t s n μ- = The higher the degrees of freedom (df) are, the narrower the spread of the t distribution As the df increase, the t density curve approaches the N (0, 1) curve more closely Generally it is more spread than the normal, especially if the df are small df = n 2 df = n 1 n 1 < n 2 The t-test Statistic and Distribution One-sample t Confidence Interval Suppose a SRS of size n is drawn from a population having unknown mean μ . A level C confidence interval for μ is Here t* is the value for the t density curve with the df = n-1. The area between – t* and t* is C The interval is exact for the normal population and approximately correct for large n in other cases Note the standard error in the formula * * * , or , - + s s s x t x t x t n n n Example 1 A mutual fund is trying to estimate the return on investment in companies that won quality awards last year. A random sample of 20 such companies is selected, and the return on investment is calculated. The mean of the sample is 14.75 and the standard deviation of the sample is 8.18. Construct a 95% confidence interval for the mean return on investment. Example 2 From a running production of corn soy blend...
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501_Lecture_07 - Section 7.1 Inference for the mean of a...

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