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Math 10 Chapter 9 Notes: Hypothesis Testing done constantly in medicine, business, polling, education, etc. •to do: set up 2 contradictory statements •first statement - often the accepted belief •conduct a test to see whether our data supports or does not support the first hypothesis • Math 10 Chapter 9 Notes: Hypothesis Testing Null hypothesis: Ho: Alternate hypothesis: Ha: Example: Ho: John loves Marcia Ha: John does not love Marcia Example: Ho: m = 6 Ha: m < 6 Math 10 Chapter 9 Notes: Hypothesis Testing Decision: Ho is: True False Do not reject Ho Correct Type II Decision Error Reject Ho Type I Error Correct Decision Math 10 Chapter 9 Notes: Hypothesis Testing Type I error: Reject the null hypothesis when the null is TRUE. P(Type I error) = a Type II error: Do not reject the null hypothesis when the null is FALSE. P(Type II error) = b Goal: Minimize a and b. Math 10 Chapter 9 Notes: Hypothesis Testing “ = , , or ” are ALWAYS in the null hypothesis Ho. “ , > , or < ” are ALWAYS in the alternate hypothesis Ha. Math 10 Chapter 9 Notes: Hypothesis Testing Examples: State the null hypothesis, Ho, and the alternative hypothesis, Ha, in terms of the appropriate parameter (m or p). At most 60% of Americans vote in presidential elections. (Right-Tailed) Ho: p 0.60 Ha: p > 0.60 Math 10 Chapter 9 Notes: Hypothesis Testing Fewer than 5% of adults ride the bus to work in New York City. (Left-tailed) Ho: p 0.05 Ha: p < 0.05 Math 10 Chapter 9 Notes: Hypothesis Testing The average number of cars a person owns in his/her lifetime is not more than 10. (Right-tailed) Ho: m 10 Ha: m > 10 Math 10 Chapter 9 Notes: Hypothesis Testing Europeans have an average paid vacation each year of six weeks. (Twotailed) Ho: m = 10 Ha: m 10 Math 10 Chapter 9 Notes: Hypothesis Testing Private universities cost, on average, more than $20,000 per year for tuition, room, and board. (Right-tailed) Ho: m 20,000 Ha: m > 20,000 Math 10 Chapter 9 Notes: Hypothesis Testing What are Type I and Type II errors for some of these problems? Ho: m 20,000 Ha: m > 20,000 Type I: We believe that private universities cost, on average, more than $20,000 per year for tuition, room, and board when, in fact, the average cost is no more than $20,000. Math 10 Chapter 9 Notes: Hypothesis Testing Ho: m 20,000 Ha: m > 20,000 Type II: We believe that private universities cost, on average, no more than $20,000 per year for tuition, room, and board when, in fact, the average cost is more than $20,000. no more than = at most = less than or equal to Math 10 Chapter 9 Notes: Hypothesis Testing To perform a hypothesis test: sample data is gathered data typically favors one of the hypotheses Decisions if data favors the null hypothesis (Ho), we “do not reject the null” if data favors the alternate hypothesis (Ha), we “reject the null” Math 10 Chapter 9 Notes: Hypothesis Testing NOTE: We are ALWAYS testing the null hypothesis, never the alternate. Our conclusion is ALWAYS in regards to the null (Ho). after a decision is made, an appropriate CONCLUSION is made regarding the null hypothesis Math 10 Chapter 9 Notes: Hypothesis Testing sometimes, data favors neither hypothesis (this implies an “inconclusive” test result) a test may be “left-tailed”, “right-tailed”, or “two-tailed” depending upon the null hypothesis associated with the null hypothesis is a “pre-conceived” a. P(Type I error ) = a Math 10 Chapter 9 Notes: Hypothesis Testing If no pre-conceived a is given, it is common practice to use a = 0.05. A test may be “left-tailed”, “right-tailed”, or “two-tailed” depending upon the null hypothesis data is collected to calculate what is called the p-value, or level of significance, or calculated a Math 10 Chapter 9 Notes: Hypothesis Testing p-value = P(the information/data will happen purely by chance GIVEN that the null hypothes is true) decision to reject or to not reject the null is based upon whether a > p-value or a < p-value Reject Ho if a > p-value Do not Reject Ho if a < p-value Math 10 Chapter 9 Notes: Hypothesis Testing Random Variable: Xbar = average … P’ = proportion … Examples: Xbar = the average tuition for a private college P’ = the proportion of voters who voted for the winning candidate