Hypothesis Testing for Proportions An example of a one sample test for proportions We could run a test of hypothesis to see if our data for Red M M s actually agrees with the advertised amount In our experiment we had an average of 10 7 or 19 Red Variable Obs Mean Std Dev Min Max Red 83 10 73494 3 679489 3 22 pRed 83 1916867 0651044 05 38 Total 83 56 06024 2 96051 44 63 Using the Steps in Hypothesis Testing 1 State H0 it ALWAYS has the and HA it s sign depends on the question asked The null hypothesis is the status quo so here it would be M M s advertised percent According to their web site this is 20 Since we just want to check this we can test H0 Red 0 20 vs HA Red 0 20 2 Determine the appropriate level depending on the consequence of Type I and II errors We ll discuss Type I and II errors later For now let s use the usual 5 This means if our observed proportion would happen less than 5 of the time if the true proportion is 20 then we re going to claim the advertised amount is NOT 20 3 Determine the appropriate test and calculate a p value use Labs Calculating Tests of Hypotheses and the flowchart to determine which Case So far we ve only talked about z tests we converted the sample proportion p to a z score and then found the probability using the Z Table There are many other types of tests that we will discuss soon For this particular type of data we will be using Case 6 the normal approximation for proportions NOTE You must verify if the Conditions for the normal approximation hold before running this test however We did this on HW 4 5 Case 6 says we need to give n and p So we put 0 05 in the box labelled alpha 56 in the n box 0 19 in the p box 0 20 in the hyp value hypothesized value the number in H0 box and finally click on Two sided in the Test box You just ignore the rest of the boxes because it isn t used The output and graph are Two Sided Test for 0 1 proportion pi approximate alpha 05 Hypothesized value 2 n 56 p 19 Z calc 18708287 Critical values 1 959964 1 959964 Fail to reject H 0 p value 85159566 4 State the conclusion if p value reject H0 otherwise fail to reject in terms of the hypothesis answer the question asked The p value is given in the output see the last line and in the last line of the title of the graph Since the p value 0 852 which is NOT 0 05 we cannot reject our null hypothesis In other words our data is quite consistent with the advertised percent of Red M M s We cannot refute their claim Another example of a one sample test for proportions What would happen if we only had one bag of M M s and it was the bag of 60 with only 5 Red Is this too few Red s Are there really less Red M M s than the advertised amount Still using the Steps in Hypothesis Testing 1 State H0 it ALWAYS has the and HA it s sign depends on the question asked Again the null hypothesis is the advertised percent 20 Now however we want to know if the true proportion is really less than the stated amount so we should test H0 Red 0 20 vs HA Red 0 20 2 Determine the appropriate level depending on the consequence of Type I and II errors Let s stay with the standard 5 level 3 Determine the appropriate test and calculate a p value use Labs Calculating Tests of Hypotheses and the flowchart to determine which Case For this data we will again use Case 6 the normal approximation for proportions NOTE 5 out of 60 is barely the necessary amount Case 6 says we need to give n and p So we put 0 05 in the box labelled alpha 60 in the n box 0 083 in the p box 0 20 in the hyp value hypothesized value the number in H0 box and finally click on Left sided in the Test box The output and graph are Left Sided Test for 0 1 proportion pi approximate alpha 05 Hypothesized value 2 n 60 p 083 Z calc 2 2656953 Critical value 1 6448536 Reject H 0 p value 01173502 4 State the conclusion if p value reject H0 otherwise fail to reject in terms of the hypothesis answer the question asked Since the p value 0 012 which IS 0 05 we reject our null hypothesis and state that there is sufficient evidence to conclude that the true proportion of Red M M s Red is actually LESS than 20 We can say that Red is statisitically significantly less than 20 In other words our data disagrees with the advertised amount enough to dispute M M s claim Why is there a difference First you need to think of what the significance level means and what a hypothesis test actually does Remember we said there was a distribution of Red M M s or the proportion of Red M M s in a regular size bag The significance level 5 means that we will be throwing out 5 of this distribution and therefore WRONG 5 of the time In hypothesis testing we assume that the center of our normal curve is the hypothesized value here it s 20 and calculate where our data falls on this curve We just happened to get a bag out there on the tail Look at the interpretation of the p value If the true proportion of Red M M s Red is actually 20 we would see 8 3 or less Red M M s only 1 2 of the time This sample wouldn t happen very often but it is still possible An example of a two sample test for proportions We could also test whether the proportions of Blue and Orange are the same as M M claims Our sample data says that there are only 4 9 Blue M M s and 6 6 Orange on average Does this means the true proportions are different Blue Orange pBlue pOrange 83 83 83 83 4 903614 6 566265 0872289 1177108 2 588033 3 151433 0438488 055619 0 0 0 0 14 14 24 26 Using the Steps in Hypothesis Testing 1 State H0 it ALWAYS has the and HA it s sign depends on the question asked We don t really care what the true proportions are we just want to see if they are the same if there really ARE less Blue M M s than Orange ones So we test H0 Blue Orange vs HA Blue Orange 2 Determine the appropriate level depending on the consequence of Type I and II errors Again we ll the usual 5 3 Determine …