Chapter 11 Two-Sample Hypothesis Tests

1. Section 11.1 Hypothesis Tests for the Differenece Between Two Means: Independent Samples
   • Notation
     • `mu_1` and `mu_2` are population means
     • `barx_1` and `barx_2` are sample means
     • `s_1` and `s_2` are sample standard deviations
     • `n_1` and `n_2` are sample sizes
   • The null hypothesis: `H_0: mu_1 = mu_2`
   • The alternate hypothesis will be one of
     • `H_1: mu_1 < mu_2`
     • `H_1: mu_1 > mu_2`
     • `H_1: mu_1 != mu_2`
   • The standard error of `barx_1 - barx_2 = sqrt((s_1^2)/(n_1) + (s_2^2)/(n_2))`
   • The test statistic is

     `t = ( (barx_1 - barx_2) - (mu_1 - mu_2) )/ sqrt((s_1^2)/(n_1) + (s_2^2)/(n_2))`

     as before, take degrees of freedom as the smaller of `n_1-1` or `n_2-1`, or use that complex formula from Chapter 10)

   • Assumptions
     • Simple random samples
     • Samples are independent
     • Each sample size is large (n>30) or its population is approximately normal
   • When the two population standard deviations are known, the the test statistic is

     `z = ( (barx_1 - barx_2) - (mu_1 - mu_2) )/ sqrt((sigma_1^2)/(n_1) + (sigma_2^2)/(n_2))` and we use the standard normal distribution.

2. Section 11.2 Hypothesis Tests for the Difference Between Two Proportions
   • Notation
     • `p_1` and `p_2` are the proportion of the category in the two populations
     • `hatp_1` and `hatp_2` are the proportion of the category in the two samples
     • `x_1` and `x_2` are the numbers of "individuals" in the category in the two samples
     • `n_1` and `n_2` are sample sizes
   • The null hypothesis: `H_0: p_1 = p_2`
   • The alternate hypothesis will be one of
     • `H_1: p_1 < p_2`
     • `H_1: p_1 > p_2`
     • `H_1: p_1 != p_2`
   • pooled proportion: `hatp = (x_1 + x_2) / (n_1 + n_2)`
   • Standard error = ` sqrt(( hatp*(1-hatp) )/(n_1) + ( hatp*(1-hatp) )/(n_2) ) = sqrt( hatp*(1-hatp)*(1/n_1 +1/n_2))`
   • Test statistic is `z = (hatp_1-hatp_2)/sqrt( hatp*(1-hatp)*(1/n_1 +1/n_2))`
   • Assumptions
     • two random samples
     • samples are independent
     • each population is at least 20 times as large as the sample drawn from it
     • we have the same two categories in each sample
     • The sample sizes are large enough so that we have at least 10 "individuals" in each category in each sample
3. Section 11.3 Hypothesis Tests for the Difference Between Two Means: Paired Samples
   • Notation
     • `bard` is the sample mean of the difference in the matched pairs
     • `s_d` is the sample standard deviation for the difference in the matched pairs
     • `mu_d` is the population mean difference for the matched pairs
   • Assumptions
     • simple random samples of matched pairs
     • large sample size (n>30) or the difference valeus have no outliers and are not skewed
   • Test statistic: `t = (bard -mu_0)/( s_d/sqrt(n) )`, with `n-1` degrees of freedom
4. Section 11.4 Hypothesis Tests for Two Population Standard Deviations
   • The `F` distribution
   • Assumptions
     • independent random samples from two populations
     • both populations are normal
   • Test statistic: `F = (max((s_1)^2,(s_2)^2))/(min((s_1)^2,(s_2)^2))`
   • `F` has two values for "degrees of freedom" and we use both of them
     • degrees of freedom for numerator = size of numerator sample minus 1
     • degrees of freedm for denominator = size of denominator sample minus 1
     • specified as `F_(ndf,ddf)` with `ndf`, the numerator degrees of freedom always given first.
   • Here is a detailed explanation of using the F test.
5. Section 11.5 The Multiple Testing Problem
   • You can only pull the trigger so many times
   • The Bonferroni Method
6. Sample problems on the calculator