Two Populations Hypothesis Testing

Transcription

1 Two Populations Hypothesis Testing Two Proportions (Large Independent Samples) Two samples are said to be independent if the data from the first sample is not connected to the data from the second sample. If the two data sets are connected, then the samples are said to be dependent. Dependent samples are also referred to as paired samples or matched samples. p 1 = x 1 n 1 p = x n To compare the two populations, we use the difference between the two sample proportions (point estimate): p 1 p 1. The two samples must be independent. The samples must be large enough to use the normal distribution. The products n 1 p 1 5 n 1 (1 p ) 1 5 n p 5 n (1 p ) 5 If the above conditions are met, the sampling distribution for p 1 p (the difference between the samples proportions) has a normal distribution with mean: μ p1 p = p 1 p Standard Error σ p1 p = p (1 1 p ) 1 + p (1 p ) n 1 n p 1 =population 1 proportion p =population proportion n 1 =sample size of population 1 n =sample size of population x 1 =number of successes in population 1 x =number of successes in population =sample proportion of population 1 =sample proportion of population p 1 p Spring 017

2 Confidence Interval (p 1 p ) ± (zα ) (σ p1 p ) *Note: p 1 and p estimates are used when p 1 and p are unknown. Standardized test statistic Z = (p 1 p ) (p 1 p ) p (1 p ) ( 1 n n ) p = x 1 + x n 1 + n p = n 1p 1 + n p n 1 + n Determine the decision rule: Reject Ho if z < If left tail =normsinv(α) Determine the decision rule: Reject Ho if z > If right tail =normsinv(1-α) Determine the decision rule: Reject Ho if z > If two tail test =normsinv(1 α ) Two Means (Sigma Known) Bounds x ± (zα) ( σ n ) x 1 x ± (zα ) (σ 1) n 1 + (σ ) n x 1 x is the point estimator x =sample 1 mean of sample 1 x =sample mean of sample σ 1 =population standard deviation of population 1 σ =population standard deviation of population (σ 1 ) =population variance of population 1 (σ ) =population variance of population (σ 1) + (σ ) is the standard deviation n 1 n Spring 017

3 Have to be large enough: n 1 30 and n 30 Z = (x 1 x ) (μ 1 μ ) (σ 1) + (σ ) n 1 n Two Means (Sigma Unknown) Use t-distribution when population standard deviation is not known x 1 x ± (tα ) (σ x1 x ) x 1 x is the point estimator Standard error when the population variances are equal σ x1 x = S P n 1 n Degrees of freedom n 1 + n s 1 =sample standard deviation of sample 1 s =sample standard deviation of sample (s 1 ) =sample variance of sample 1 (s ) =sample variance of sample Standard error when the population variances are not equal σ x1 x = (s 1) + (s ) n 1 n Degrees of freedom is the smaller of n 1 1 and n 1 Pooled sample standard deviation S P = (n 1 1)(s 1 ) + (n 1)(s ) n 1 + n Common population variance S P = (n 1 1)(s 1 ) + (n 1)(s ) n 1 + n Spring 017

4 Find tα α by looking up in table depending on the degrees of freedom and t = (x 1 x ) (μ 1 μ ) σ x1 x *μ 1 μ = 0 Decision Rule: Reject Ho if t > If right tail: look up α and degrees of freedom Reject Ho if t > If two tail: look up degrees of freedom and α Two Means (Dependent Samples) When dependent samples are involved, the data is thought of as paired data. Difference between the pairs of data values is referred to as a paired difference. Paired difference d = x 1 x The difference between two population means, when dependent samples are used, is equivalent to the mean of the paired differences. If the following two conditions are met, the sampling distribution for d, the mean of the differences of the paired data entries in the dependent samples, has a t- distribution with n-1 degrees of freedom, where n is the number of data pairs. 1. the two samples must be dependent (paired). each population has a normal distribution Mean of the population μ d = μ 1 μ Point estimate/mean of the paired differences d = d n n = number of data pairs Spring 017

5 Bounds d ± (tα) ( s d n ) Sample standard deviation s d = n( d ) ( d) n(n 1) Test statistic t = d μ d ( s d n ) n 1= degrees of freedom Decision Rule: Reject Ho if t > If right tail: look up n-1 degrees of freedom and α Decision Rule: Reject Ho if t < If left tail: look up n-1 degrees of freedom and α and make it negative Decision Rule: Reject Ho if t > If two tail: look up n-1 degrees of freedom and α Two Population Variances In order to compare two population variances, must ensure that the following two conditions are met: 1. the two populations must be independent, not matched or paired in any way, and. the two populations must be normally distributed F-distribution Skewed to the right Values of F are always greater than 0 Shape is completely determined by its two parameters, the degrees of freedom of the numerator and the degrees of freedom of the dominator of the ratio Spring 017

6 Degrees of freedom of numerator = n 1 1 Degrees of freedom of denominator = n 1 Ho/Ha: σ 1 F = s 1 s If left tail: Decision Rule: Reject Ho if F F 1 α If right tail: Decision Rule: Reject Ho if F F α If two tail: Decision Rule: Reject Ho if F F 1 α OR Reject Ho if F Fα n 1 =sample size of sample 1 n =sample size of sample σ 1 =population standard deviation of population 1 σ =population standard deviation of population (σ 1 ) =population variance of population 1 (σ ) =population variance of population s 1 =sample standard deviation of sample 1 s =sample standard deviation of sample s 1 =sample variance of sample 1 s =sample variance of sample Spring 017