Statistics Part IV Confidence Limits and Hypothesis Testing. Joe Nahas University of Notre Dame

Transcription

1 Statistics Part IV Confidence Limits and Hypothesis Testing Joe Nahas University of Notre Dame

2 Statistic Outline (cont.) 3. Graphical Display of Data A. Histogram B. Box Plot C. Normal Probability Plot D. Scatter Plot E. MatLab Plotting 4. Confidence Limits and Hypothesis Testing A. Student s t Distribution i. Who is Student ii. Definitions B. Confidence Limits for the Mean C. Equivalence of two Means D. Equivalence of two Variances 2 2

3 Student s t Distribution Suppose a random sample of size n is drawn from a normal N(μ,σ) population. If x is the estimate of the mean from the sample, and s is the sample standard deviation, then t = x μ s / n has the t distribution with n 1 degrees of freedom. There is a different t distribution for each sample size n as specified in the degrees of freedom. WE ARE NOT GOING TO PROVE THIS! NIST ESH

4 Who was Student? The t distribution was discovered by William S. Gossett, a statistician employed by the Guinness Brewing Company. He was trying to determine how accurate the data from his small samples were. Guinness previously had problems with proprietary information being published so it required Gossett not to publish his discoveries under his own name. Guinness did not want its competitors to know that it was using statistics to improve its beer. He published the t distribution under the pen name Student in The distribution is usually referred to as Student s t Distribution 4

5 Student s t distribution The probability density function for the t distribution is: where B is the Beta function and ν is a positive integer shape parameter. The Beta function is: f (x) = Β(α,β) = 1 + x 2 ν Β(0.5,0.5ν) ν 1 0 (ν +1) 2 t α 1 (1 t) β 1 dt The t distribution is equal to the Cauchy distribution for ν = 1. The t distribution approaches the normal distribution for large ν. 5

6 t Probability Density Function Large tails for ν = 1 Approaches Normal Distribution for large ν 6

7 Confidence Limits for the Mean By definition t = x μ s / n So s μ = x t α,ν n The probably value of μ is distributed around x. NIST ESH

8 t distribution Table Instructions NIST ESH

9 t distribution table NIST ESH

10 t distribution table NIST ESH

11 Confidence Limits of the Mean Confidence Limits are a two sided test. i.e. the real mean can be greater than or less then the estimate. Example: n = 195 m = s = % confidence interval α = 0.05 t 1 α/2,194 = Lower Limit = m t * s / sqrt(n) = * / sqrt(195) = Upper Limit = m + t * s / sqrt(n) = * / sqrt(195) = % of area 95% of area 2.5% of area 11

12 t distribution in Excel Values of t can be obtained using the TINV function in Excel. TINV(probability, degrees of freedom) probability = α for a two sided distribution e.g. instead of 1 α/2 = in table, use α = 0.05 degrees of freedom = ν = n 1 For the previous example: =tinv(0.05, 194) returns For a one sided distribution, use 2*α Hint: Before using tinv, try duplicating an example in the NIST ESH. 12

13 Are two Means Possibly Equal We have two estimates of the mean, m 1 and m 2 with m 1 > m 2. We have two estimates of the standard deviation, s 1 and s 2. We have two sample sizes, n 1 and n 2. This is a one sided test. Null Hypothesis: μ 1 = μ 2. Test Statistic x T = 1 x 2 s 2 1 / n 1 + s 2 2 / n 2 Similar to t = x μ s / n NIST ESH

14 Are two Means Possibly Equal Reject the Null Hypothesis if: where t 1 α,ν is the critical value of the t distribution with ν degrees of freedom where ν = T > t 1 α,ν (s 1 2 / n 1 + s 2 2 / n 2 ) 2 (s 1 2 / n 1 ) 2 /(n 1 1) + (s 2 2 / n 2 ) 2 /(n 2 1) 14

15 Equal Variances If equal variances are assumed: T = x 1 x 2 s p 1/n 1 + 1/n 2 and where s p is the pooled estimate of the standard deviation: s p = (n 1 1)s (n 2 1)s 2 2 n 1 + n 2 2 ν = n 1 + n

16 Example Mileage Data from US and Japanese cars in 1990s n 1 = 79 m 1 = s 1 = n 2 = 249 m 2 = s 2 = Assuming variances are equal T = s p = ν = 326 For 95% confidence, α = 0.05 t 0.95,ν=326 = Since T > t, the hypothesis that the means are equal is rejected! 95% of area 5 % of area 16

17 Large n What happens to the confidence limits as n gets large? μ = lim x (x t α,ν sn ) = x 17

18 Are Two Variances Equal? Null Hypothesis, H0: σ 12 = σ 2 2 Alternative Hypotheses, Ha: σ 12 < σ 2 2 for a lower one tailed test σ 12 > σ 2 2 for an upper one tailed test σ 12 σ 2 2 for a two tailed test Test Statistic: F = s 12 /s 2 2 Where s 12 and s 22 are the sample variances with sample sizes of N 1 and N 2 respectively Significance Level: α NIST ESH

19 Are Two Variances Equal? (cont.) The Hypothesis that the two variances, σ12, and σ22, are equal is rejected if: for an upper one tailed test F > F α, N1 1,N 2 1 F < F 1 α, N1 1,N 2 1 for a lower one tailed test F > F α, N1 1,N 2 1 or for a two tailed test F α, N1 1,N 2 1 F < F 1 α, N1 1,N 2 1 where is the critical value of the F distribution with N 1 1 and N 2 1 degrees of freedom and a significance level of α. NIST ESH

20 F Distribution 20

21 F Distribution NIST ESH

22 Using Excel for F Dist Use finv(α, N 1, N 1) function in Excel. Use exampel in NIST ESH to check usage. 22

23 Ceramic Data Example Is 65.5 significantly different from 61.9? From Excel NIST ESH