Standard normal distribution. t-distribution, (df=5) t-distribution, (df=2) PDF created with pdffactory Pro trial version

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1 t-ditribution In biological reearch the population variance i uually unknown and an unbiaed etimate,, obtained from the ample data, ha to be ued in place of σ. The propertie of t- ditribution are: -It i ymmetrical about the mean. -It i really family ditribution, which depend on the degree of freedom (df), of ample. 3-It i not normal ditribution and it approach to normal a ( n ) approache infinity. 4-t curve i omewhat flatter than that for tandardized normal ditribution. 5-It ha a mean zero and variance more than one but it approache one a the ample ize become large. Standard normal ditribution t-ditribution, (df=5) t-ditribution, (df=) 0 3

2 Confidence Interval for Mean μ,, when σ i Unknown,, and n 30 (C.I): A (-α) 00% C.I. for i given by x ± t -α/, v v = df From percentile of the t-ditribution table contruct thi value, t α /, Example: A ample of 6 ten-year year-old girl given a mean weight of 7.5 and tandard deviation of pound, repectively. Auming normality. Find 95 percent confidence interval for, (μ i true population mean of the weight girl). A 95% C.I. for μ i, Meaning that baed on ample tatitic we are 95% confident that the true population mean of the weight girl lie within the range of to pound. n ( ) v, 7.5 ±.35 6 (65.06, )

3 Note: When the ample n > 30, our faith in () a an approximation of (σ) i uually ubtantial, and we may feel jutified in uing normal ditribution theory to contruct confidence interval. Confidence Interval for the Difference between Two Population Mean μ and μ ; When σ = σ but are unknown, (C.I); A (- α) 00% confidence interval for μ - μ i given by, ( x - x) ± t Sp, v = n n -α/, v n n - Sp = ( n ) ( n ) n n (Pooled etimate of tandard deviation)

4 Example: The following data repreent the fating blood glucoe of ample for normal male and female. Aume that the fating blood glucoe of variance are equal (σ = σ ), but unknown. Contruct 95 percent confidence interval for the difference between the two true population mean of the fating blood glucoe. Blood Glucoe (mg/dl) Male: Female: % C.I. for μ - μ i, Sp = x x = = 99.4 ( 7 - ) ( 5 ) 7 5 = = 7.76 ( x -x ) ± t Sp, v= n n - - / α, v n n,, = ( ) ± (.8)( 7.76) ( 3.83, 6.44) 7 5

5 Meaning that baed on ample tatitic we are 95% confident that the true difference between two true population mean of the two group of fating blood glucoe lie within the range of 3.83 to Confidence Interval for the Difference between Two Population Mean μ and μ ;when σ σ but they are unknown, (C.I); An approximate ( -α) 00% confidence interval for μ - μ by, or P ( ) x x ( ) ( ) x x t μ μ x x α/, v n df = v = ± t / α, v n n n ( /n /n) ( /n ) /( n ) /n t α/, v [ ] [( ) ( )] / n i given = α n n

6 Example: Total erum complement activity (CH 50 ) wa aayed in 0 apparently healthy ubject and 0 ubject with dieae. The following reult were obtained: Subject n x With dieae Normal The invetigator had reaon to believe that the ampled population are approximately normally ditributed, but they were unwilling to aume that the two unknown population variance are equal. Find 95 percent confidence interval for μ - μ. An approximate 95% C.I. for μ - μ. v = = ± ± 4.4 ( 9.0, 39.8 ) Meaning that baed on ample tatitic we are 95% confident that the true difference between two true population mean of the two group lie within the range of 9.0 to 39.8.

7 Percentile of the Student' t ditribution df Probability that a larger value of t would be oberved = minu the value hown

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