BCOR 1020 Business Statistics Lecture 20 – April 3, 2008

Overview Chapter 9 – Hypothesis Testing –Problem: Testing A Hypothesis on a Proportion –Testing a Mean (  ): Population Variance (  ) Known –Problem: Testing a Mean (  ) when Population Variance (  ) is Known

Problem: Testing A Hypothesis on a Proportion Suppose we revisited our earlier example and conducted an additional survey to determine whether the proportion of our target market that is willing to pay $25 per unit exceeds 20% (as required by the business case). In our new survey, 72 out of 300 respondents said they would be willing to pay $25 per unit. At the 5% level of significance, conduct the appropriate hypothesis test to determine whether the population proportion exceeds 20%. Include the following: –State the level of significance, . –State the null and alternative hypotheses, H 0 and H 1. –Compute the test statistic –State the decision criteria –State your decision (Overhead)

Problem: Testing A Hypothesis on a Proportion Work:  = H 0 : H 1 : Test Statistic (and Distribution under H 0 )… Decision Criteria… Decision…

Clickers What are the appropriate null and alternative hypotheses? (B) H 0 :  >  H 1 :  <  (A) H 0 :  <  H 1 :  >  (C) H 0 :  =  H 1 :  

Clickers What is the point estimate of the population proportion  ? (A) p = 0.15 (B) p = 0.20 (C) p = 0.22 (D) p = 0.24 (E) p = 0.36

Clickers What is the calculated value of your test statistic? (A) Z* = 2.00 (B) Z* = 1.73 (C) Z* = 0.87 (D) Z* = 0.71

Clickers What is your decision criteria? (A) Reject H 0 if Z* < (B) Reject H 0 if Z* < (C) Reject H 0 if Z* > (D) Reject H 0 if Z* > (E) Reject H 0 if |Z*| > 1.960

Problem: Testing A Hypothesis on a Proportion Conclusion: Since our test statistic Z* = 1.73 > Z  = 1.645, we will reject H 0 in favor of H 1 :  > Based on the data in this sample, there is statistically significant evidence that the population proportion exceeds 20%.

Chapter 9 – Testing a Mean (  known) Hypothesis Tests on  (  known): If we wish to test a hypothesis about the mean of a population when  is assumed to be known, we will follow the same logic… –Specify the level of significance,  (given in problem or assume 10%). –State the null and alternative hypotheses, H 0 and H 1 (based on the problem statement). –Compute the test statistic and determine its distribution under H 0. –State the decision criteria (based on the hypotheses and distribution of the test statistic under H 0 ). –State your decision.

Chapter 9 – Testing a Mean (  known) Selection of H 0 and H 1 : Remember, the conclusion we wish to test should be stated in the alternative hypothesis. Based on the problem statement, we choose from… (i)H 0 :  >  0 H 1 :  <  0 (ii) H 0 :  <  0 H 1 :  >  0 (iii) H 0 :  =  0 H 1 :   0 where  0 is the null hypothesized value of  (based on the problem statement).

Chapter 9 – Testing a Mean (  known) Test Statistic: Start with the point estimate of , Recall that for a large enough n {n > 30}, is approximately normal with and If H 0 is true, then is approximately normal with and So, the following statistic will have approximately a standard normal distribution:

Chapter 9 – Testing a Mean (  known) Decision Criteria: Just as with tests on proportions, our decision criteria will consist of comparing our test statistic to an appropriate critical point in the standard normal distribution (the distribution of Z* under H 0 ). (i)For the hypothesis test H 0 :  >  0 vs. H 1 :  <  0, we will reject H 0 in favor of H 1 if Z* < – Z . (ii) For the hypothesis test H 0 :   0, we will reject H 0 in favor of H 1 if Z* > Z . (iii) For the hypothesis test H 0 :  =  0 vs. H 1 :   0, we will reject H 0 in favor of H 1 if |Z*| > Z  /2.

Chapter 9 – Testing a Mean (  known) Example (original motivating example): Suppose your business is planning on bringing a new product to market. There is a business case to proceed only if –the cost of production is less than $10 per unit and –At least 20% of your target market is willing to pay $25 per unit to purchase this product. How do you determine whether or not to proceed?

Chapter 9 – Testing a Mean (  known) Motivating Example (continued) : Assume the cost of production can be modeled as a continuous variable. –You can conduct a random sample of the manufacturing process and collect cost data. –If 40 randomly selected production runs yield and average cost of $9.00 with a standard deviation of $1.00, what can you conclude? –We will test an appropriate hypothesis to determine whether the average cost of production is less than $10.00 per unit (as required by the business case). (Overhead)

Chapter 9 – Testing a Mean (  known) Motivating Example (continued) : Conduct the Hypothesis Test: –Specify , say  = 0.05 for example. –Select appropriate hypotheses. Since we want to test that the mean is less than $10, we know that  0 = 10 and H 1 should be the “<“ inequality. So we will test (i) H 0 :  >  0 vs. H 1 :  <  0. –Calculate the test statistic…

Chapter 9 – Testing a Mean (  known) Motivating Example (continued) : Conduct the Hypothesis Test: –Use the Decision criteria: (i) we will reject H 0 in favor of H 1 if Z* < – Z , where Z.05 = –State our Decision… Since Z* = < -Z.05 = , we reject H 0 in favor of H 1. –In “plain” language… There is statistically significant evidence that the average cost of production is less than $10 per unit.

Chapter 9 – Testing a Mean (  known) Calculating the p-value of the test: The p-value of the test is the exact probability of a type I error based on the data collected for the test. It is a measure of the plausibility of H0. –P-value = P(Reject H0 | H0 is True) based on our data. –Formula depends on which pair of hypotheses we are testing… (i)For the hypothesis test H 0 :  >  0 vs. H 1 :  <  0, (ii) For the hypothesis test H 0 :   0, (iii) For the hypothesis test H 0 :  =  0 vs. H 1 :   0,

Chapter 9 – Testing a Mean (  known) Example: Let’s calculate the p-value of the test in our example… We found Z* = –6.32 Since we were testing H 0 :  >  0 vs. H 1 :  <  0, Interpretation: If we were to reject H 0 based on the observed data, there is approximately zero probability that we would be making a type I error. Since this is smaller than  = 5%, we will reject H 0.

Problem: Testing A Hypothesis on a Mean (  Known) In an effort to get a loan, a clothing retailer has made the claim that the average daily sales at her store exceeds $7500. Historical data suggests that the purchase amount is normally distributed with a standard deviation of  = $1500. In a randomly selected sample of 24 days sales data, the average daily sales were found to be = $7900. At the 10% level of significance, conduct the appropriate hypothesis test to determine whether the data supports the retailer’s claim. Include the following: –State the level of significance, . –State the null and alternative hypotheses, H 0 and H 1. –Compute the test statistic –State the decision criteria –State your decision (Overhead)

Problem: Testing A Hypothesis on a Mean (  Known) Work:  = H 0 : H 1 : Test Statistic (and Distribution under H 0 )… Decision Criteria… Decision…

Clickers What are the appropriate null and alternative hypotheses? (A) H 0 :  >  H 1 :  <  (B) H 0 :  <  H 1 :  >  (C) H 0 :  =  H 1 :  

Clickers What is the calculated value of your test statistic? (A) Z* = 0.27 (B) Z* = 1.28 (C) Z* = 1.31 (D) Z* = 1.96 (E) Z* = 5.33

Clickers What is your decision criteria? (A) Reject H 0 if Z* < (B) Reject H 0 if Z* < (C) Reject H 0 if Z* > (D) Reject H 0 if Z* > (E) Reject H 0 if |Z*| > 1.960

Clickers What is your decision? (A) Reject H 0 in favor of H 1 (B) Fail to reject (Accept) H 0 in favor of H 1 (C) There is not enough information

Problem: Testing A Hypothesis on a Mean (  Known) Conclusion: Since our test statistic Z* = 1.31 is greater than Z  = 1.282, we will reject H 0 in favor of H 1 :  > $7500. Based on the data in this sample, there is statistically significant evidence that the average daily sales at her store exceeds $7500.

Clickers Given our test statistics Z* = 1.31, calculate the p- value for the hypothesis test H 0 :  < 7500 vs. H 1 :  > (A) p-value = (B) p-value = (C) p-value = (D) p-value =