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BCOR 1020 Business Statistics

BCOR 1020 Business Statistics. Lecture 20 – April 3, 2008. Overview. Chapter 9 – Hypothesis Testing Problem: Testing A Hypothesis on a Proportion Testing a Mean ( m ): Population Variance ( s ) Known Problem: Testing a Mean ( m ) when Population Variance ( s ) is Known.

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BCOR 1020 Business Statistics

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  1. BCOR 1020Business Statistics Lecture 20 – April 3, 2008

  2. Overview • Chapter 9 – Hypothesis Testing • Problem: Testing A Hypothesis on a Proportion • Testing a Mean (m): Population Variance (s) Known • Problem: Testing a Mean (m) when Population Variance (s) is Known

  3. 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, a. • State the null and alternative hypotheses, H0 and H1. • Compute the test statistic • State the decision criteria • State your decision (Overhead)

  4. Problem: Testing A Hypothesis on a Proportion Work: a = H0: H1: Test Statistic (and Distribution under H0)… Decision Criteria… Decision…

  5. Clickers What are the appropriate null and alternative hypotheses? (A) H0: p<0.20 H1: p > 0.20 (B) H0: p>0.20 H1: p < 0.20 (C) H0: p = 0.20 H1: p0.20

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

  7. 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

  8. Clickers What is your decision criteria? (A) Reject H0 if Z* < -1.645 (B) Reject H0 if Z* < -1.960 (C) Reject H0 if Z* > 1.960 (D) Reject H0 if Z* > 1.645 (E) Reject H0 if |Z*| > 1.960

  9. Problem: Testing A Hypothesis on a Proportion Conclusion: • Since our test statistic Z* = 1.73 > Za = 1.645, we will reject H0 in favor of H1: p > 0.20. • Based on the data in this sample, there is statistically significant evidence that the population proportion exceeds 20%.

  10. Chapter 9 – Testing a Mean (s known) Hypothesis Tests on m (s known): • If we wish to test a hypothesis about the mean of a population when s is assumed to be known, we will follow the same logic… • Specify the level of significance, a (given in problem or assume 10%). • State the null and alternative hypotheses, H0 and H1 (based on the problem statement). • Compute the test statistic and determine its distribution under H0. • State the decision criteria (based on the hypotheses and distribution of the test statistic under H0). • State your decision.

  11. Chapter 9 – Testing a Mean (s known) Selection of H0 and H1: • Remember, the conclusion we wish to test should be stated in the alternative hypothesis. • Based on the problem statement, we choose from… • H0: m>m0 • H1: m < m0 (ii) H0: m<m0 H1: m > m0 (iii) H0: m = m0 H1: mm0 where m0 is the null hypothesized value of m (based on the problem statement).

  12. Chapter 9 – Testing a Mean (s known) Test Statistic: • Start with the point estimate of m, • Recall that for a large enough n {n> 30}, is approximately normal with and • If H0 is true, then is approximately normal with and • So, the following statistic will have approximately a standard normal distribution:

  13. Chapter 9 – Testing a Mean (s 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 H0). • For the hypothesis test H0: m>m0 vs. H1: m < m0, we will reject H0 in favor of H1 if Z* < – Za. (ii) For the hypothesis test H0: m<m0 vs. H1: m > m0, we will reject H0 in favor of H1 if Z* > Za. (iii) For the hypothesis test H0: m = m0 vs. H1: mm0, we will reject H0 in favor of H1 if |Z*| > Za/2.

  14. Chapter 9 – Testing a Mean (s 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?

  15. Chapter 9 – Testing a Mean (s 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)

  16. Chapter 9 – Testing a Mean (s known) Motivating Example (continued) : • Conduct the Hypothesis Test: • Specify a, say a = 0.05 for example. • Select appropriate hypotheses. • Since we want to test that the mean is less than $10, we know that m0 = 10 and H1 should be the “<“ inequality. • So we will test (i) H0: m>m0 vs. H1: m < m0. • Calculate the test statistic…

  17. Chapter 9 – Testing a Mean (s known) • Motivating Example (continued) : • Conduct the Hypothesis Test: • Use the Decision criteria: • (i) we will reject H0 in favor of H1 if Z* < – Za, where Z.05 = 1.645. • State our Decision… • Since Z* = -6.32 < -Z.05 = -1.645, we reject H0 in favor of H1. • In “plain” language… • There is statistically significant evidence that the average cost of production is less than $10 per unit.

  18. 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… Chapter 9 – Testing a Mean (s known) • For the hypothesis test H0: m>m0 vs. H1: m < m0, (ii) For the hypothesis test H0: m<m0 vs. H1: m > m0, (iii) For the hypothesis test H0: m = m0 vs. H1: mm0,

  19. Example: Let’s calculate the p-value of the test in our example… We found Z* = –6.32 Since we were testing H0: m>m0 vs. H1: m < m0, Chapter 9 – Testing a Mean (s known) Interpretation: If we were to reject H0 based on the observed data, there is approximately zero probability that we would be making a type I error. Since this is smaller than a = 5%, we will reject H0.

  20. Problem: Testing A Hypothesis on a Mean (s 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 s = $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, a. • State the null and alternative hypotheses, H0 and H1. • Compute the test statistic • State the decision criteria • State your decision (Overhead)

  21. Problem: Testing A Hypothesis on a Mean (s Known) Work: a = H0: H1: Test Statistic (and Distribution under H0)… Decision Criteria… Decision…

  22. Clickers What are the appropriate null and alternative hypotheses? (A) H0: m>7500 H1: m < 7500 (B) H0: m<7500 H1: m > 7500 (C) H0: m = 7500 H1: m7500

  23. 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

  24. Clickers What is your decision criteria? (A) Reject H0 if Z* < -1.645 (B) Reject H0 if Z* < -1.282 (C) Reject H0 if Z* > 1.282 (D) Reject H0 if Z* > 1.645 (E) Reject H0 if |Z*| > 1.960

  25. Clickers What is your decision? (A) Reject H0 in favor of H1 (B) Fail to reject (Accept) H0 in favor of H1 (C) There is not enough information

  26. Problem: Testing A Hypothesis on a Mean (s Known) • Conclusion: • Since our test statistic Z* = 1.31 is greater than Za = 1.282, we will reject H0 in favor of H1: m > $7500. • Based on the data in this sample, there is statistically significant evidence that the average daily sales at her store exceeds $7500.

  27. Clickers Given our test statistics Z* = 1.31, calculate the p- value for the hypothesis test H0: m< 7500 vs. H1: m > 7500. (A) p-value = 0.0060 (B) p-value = 0.0951 (C) p-value = 0.9940 (D) p-value = 0.9049

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