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Section 7.4-7.5

# Section 7.4-7.5 - Sections 7.4-7.5 7.4 Small Sample...

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Unformatted text preview: Sections 7.4-7.5 7.4 Small Sample Intervals Based on a Normal Population Distribution • t-distribution • confidence intervals, upper and lower bounds for a small sample distribution • prediction intervals We know from CLT that 1. If n is large , then for any population s n x z ) ( μ- = is approximately standard normal, and that 2. If population is normal and standard deviation σ is known then for any n the sampling distribution of x is normally distributed and σ μ n x z ) (- = has standard normal distribution. However, if in the formula above we replace σ by s and n is small, then the distribution of the above statistic can be far from 1 standard normal. We introduce a new probability distribution for a small sample. t-distribution Definition : Let x 1 , ... x n be a random sample from a normal distribution.. Then the standardized variable Has a so called t-distribution with n-1 degrees of freedom Properties of t-curves: • Symmetric and centered at zero • Thicker tails if compared to z-curve • Different for different n (n-1 called degrees of freedom) • When n → ∞, the t-curves approach z-curve 2 n s x t μ- = Example 1: Find the t- critical value for a) Central area = .95 for df = 10 b) Upper tail area = .01, df = 25 c) Lower tail area = .025, df = 5 Solution : (a) From Table IV ( row 10, central area = .95), the critical value is 2.228. (b) An upper tail area of .01 corresponds to a cumulative area of .99, so from Table IV ( row 25, cumulative area area = .99), the critical value is 2.485. (c) A lower tail area of .025 corresponds to a central area of .95, so from Table IV ( row 5, central area = .95), the critical value is 2.571. One Sample t Confidence Interval for μ Assumptions: • Random sample from normal population ) , ( σ μ N No assumption about n 3 The two-sided CI for μ is of the form n s t x c ± , where c t is the central area critical value for t distribution with df = n-1 to be found in Table IV. )- (1 100 α % Upper Confidence Bound: n s t x c 1 + < μ where 1 c t is t-critical value for cumulative (one sided) area )- (1 100 α % Lower Confidence Bound: n s t x c 1- μ , where )- 1 ( 1 1 α = c t c is t-critical value for cumulative (one sided) area. Exercise 39: An article contained the following observations on degree of polymerization for paper specimens for which viscosity times concentration fell in a certain middle range: 418 412 421 422 425 427 431 434 437 439 446 447 448 453 454 463 465 a) Construct a boxplot of the data and comment on any interesting features. 4 1-1-2 470 460 450 440 430 420 No rm a l qua ntile b) Is it plausible that the given sample observations were selected from a normal distribution?...
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Section 7.4-7.5 - Sections 7.4-7.5 7.4 Small Sample...

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