# Line 1: a small demonstration of hypothesis testing, # in this case for a population proportion # # First, we will get a population # In this case we will get a large population source("../gnrnd5.R") gnrnd5(34981412407,967845) #let us look at the head and tail values head(L1,40) tail(L1,40) min(L1) max(L1) # ########################## # ## Problem: test the null hypothesis # ## that the population proportion of the # ## value 5 is 0.20 # ## against the alternative hypothesis that # ## the population proportion of the value # ## 5 is greater than 0.20. # ## Run the test at the 0.05 level of significance. # ########################## # take a simple random sample of size 90 # # Be careful: Every time we do this we get # a different random sample # L2 <- as.integer( runif(90, 1, 4126) ) # L2 holds the index values of our simple random sample L2 L3 <- L1[ L2 ] # L3 holds the simple random sample L3 # we need to find the proportion of 5's in the # sample table( L3 ) num_5s <- table(L3)[5] num_5s n <- length( L3 ) phat <- num_5s / n phat # # using the null hypothesis that p=0.20 find # we note that the n*p > 10 and n*(1-p)>10 # so we can use the normal approximation ... # # find the standard deviation of sample proportions sdsp <- sqrt( 0.20*(1-0.20)/n) sdsp # # find the z value that has 5% of the area to its right z <- qnorm(0.05, lower.tail=FALSE) z # The critical value is # p + z*sdsp 0.20 + z * sdsp # reject the null hypothesis if the sample proportion # is greater than that value # # alternatively we could use the attained significance # approach. How strange would it be to get the sample # proportion that we found if the true proportion is # 0.20 pnorm( phat, mean=0.20, sd= sdsp, lower.tail=FALSE) # reject if that is less than 0.05 # # Of course, we could use the function # hypoth_test_prop() to do both of these approaches in # one easy step. # source("../hypo_prop.R") hypoth_test_prop( 0.20, num_5s, 90, 1, 0.05) # # ################################# # go back and execute lines 24-77 many more times. # Each time you get a different random sample. # Keep track of the number of times that you reject or # do not reject the null hypothesis. By the way, the # true proportion of 5'2 is 0.2613333 ################################# # # now we will do the same thing for a different population # gnrnd5(34981412407,869895) #let us look at the head and tail values head(L1,40) tail(L1,40) min(L1) max(L1) # ########################## # ## Problem: test the null hypothesis # ## that the population proportion of the # ## value 5 is 0.20 # ## against the alternative hypothesis that # ## the population proportion of the value # ## 5 is greater than 0.20. # ## Run the test at the 0.05 level of significance. # ########################## # take a simple random sample of size 90 # # Be careful: Every time we do this we get # a different random sample # L2 <- as.integer( runif(90, 1, 4126) ) # L2 holds the index values of our simple random sample L2 L3 <- L1[ L2 ] # L3 holds the simple random sample L3 # we need to find the proportion of 5's in the # sample table( L3 ) num_5s <- table(L3)[5] num_5s n <- length( L3 ) phat <- num_5s / n phat # # using the null hypothesis that p=0.20 find # we note that the n*p > 10 and n*(1-p)>10 # so we can use the normal approximation ... # # find the standard deviation of sample proportions sdsp <- sqrt( 0.20*(1-0.20)/n) sdsp # # find the z value that has 5% of the area to its right z <- qnorm(0.05, lower.tail=FALSE) z # The critical value is # p + z*sdsp 0.20 + z * sdsp # reject the null hypothesis if the sample proportion # is greater than that value # # alternatively we could use the attained significance # approach. How strange would it be to get the sample # proportion that we found if the true proportion is # 0.20 pnorm( phat, mean=0.20, sd= sdsp, lower.tail=FALSE) # reject if that is less than 0.05 # # Of course, we could use the function # hypoth_test_prop() to do both of these approaches in # one easy step. # hypoth_test_prop( 0.20, num_5s, 90, 1, 0.05) # # ################################# # go back and execute lines 114-162 many more times. # Each time you get a different random sample. # Keep track of the number of times that you reject or # do not reject the null hypothesis. By the way, the # true proportion of 5'2 is 0.1968485 ################################# #