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Welcome to BUAD 310

Welcome to BUAD 310. Instructor: Kam Hamidieh Lecture 14, Monday March 10, 2014. Agenda & Announcement. Today: Finish up from last time Start Chapter 16: We start with means & then do proportions ( your book does the proportions first ) Reading for Chapters 16: ALL of it!

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Welcome to BUAD 310

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  1. Welcome to BUAD 310 Instructor: Kam Hamidieh Lecture 14, Monday March 10, 2014

  2. Agenda & Announcement • Today: • Finish up from last time • Start Chapter 16: We start with means & then do proportions ( your book does the proportions first ) • Reading for Chapters 16: ALL of it! • Homework 4 will be posted soon & due after spring break. BUAD 310 - Kam Hamidieh

  3. From Last Time • A confidence interval is a range of plausible values for a parameter of interest. • Confidence intervals: Point estimate ± margin of error • It has two main parts: • An interval of form (a, b), a < b. • A confidence level such as 95%, 99%, or 90%; • The general notation is (1-α)100% so for 95% CI is α = 5%, for 99% CI α = 1%, etc. BUAD 310 - Kam Hamidieh

  4. From Last Time • (100-α)% confidence interval for population mean μ: • Recall: 95% confident ≠ 95% probability. • Sample proportion estimates population proportion p. BUAD 310 - Kam Hamidieh

  5. Statistical Inference • Statistical inference is about drawing conclusions about a population • There are two types of approaches: • Confidence intervals • Tests of statistical significance or statistical hypothesis testing • Sometimes the word “statistical” is often dropped. BUAD 310 - Kam Hamidieh

  6. Statistical Tests • In statistics, a hypothesis is a statement about a population, usually of the form that a certain parameter takes a particular numerical value or falls in a certain range of values. • You’ll always start with two competing hypotheses: the null and the alternative. Both can not be true at the same time. • The probability question on which hypothesis testing is based: If the null hypothesis is true about the population, what is the probability of observing sample data like the one being observed? BUAD 310 - Kam Hamidieh

  7. Some Definitions • The null hypothesis, represent by the symbol H0, is the statement that there is nothing happening. It can be thought of as the status quo, or no relationship, or no difference. Often the researcher hopes to reject the null hypothesis. • The alternative hypothesis, represented by the symbol Ha, is a statement that something is happening. Often this is what the researcher supports. It may be a statement that the assumed status quo is not true, or that there is a relationship, or that there is a difference. BUAD 310 - Kam Hamidieh

  8. Example Trash Bags: a marketer of trash bags wants to use hypothesis testing to support its claim that the mean breaking strength of its new trash bag is greater than 50 pounds.μ = (Population) mean breaking strength of the new trash bags H0: μ ≤ 50 (The new trash bags are like the old ones.) Ha: μ> 50 (The new trash bags are better.) H0: μ ≤ 50 Ha: μ> 50 50 BUAD 310 - Kam Hamidieh

  9. Example Payment Time: a consulting firm wants to use hypothesis testing to see if there is evidence that the new electronic billing system has reduced the mean payment time to below 19 days.μ = (Population) mean payment time using the new billing system H0: μ ≥ 19 (New billing system is the same as before.) Ha: μ< 19 (New billing system is better.) Ha: μ< 19 H0: μ ≥ 19 19 BUAD 310 - Kam Hamidieh

  10. Example An X-ray scan machine is used to measure bone mineral density. One of the tests done to make sure the X-ray machine is working properly is to measure the mineral density of a known material called a “phantom” with density of 1.4 grams/cm2. The tester wants to scan the phantom a numbers of times and compare the mean to the correct value of 1.4 grams/cm2. Values that are too high or too low indicate that the X-ray machine is not working properly.μ = (Population) mean X-ray density reading of the phantom material H0: μ = 1.4 (Machine is working properly.) Ha: μ≠ 1.4 (Machine is not working properly.) H0: μ= 1.4 Ha: μ< 1.4 Ha: μ> 1.4 1.4 BUAD 310 - Kam Hamidieh

  11. Caution • By means of hypothesis testing one determines whether or not cases such as the ones in the previous slide are compatible with the available data. • Statistical inference does NOT lead to the proof of a hypothesis. It only indicates whether the hypothesis is or is not supported by the available data. BUAD 310 - Kam Hamidieh

  12. Examples One-Sided, Greater Than H0:  ≤ 50Ha:  > 50 (Trash Bag) One-Sided, Less Than H0:  ≥19Ha:  < 19 (Payment Time ) Two-Sided, Not Equal To H0:  = 1.4Ha: 1.4(Density) BUAD 310 - Kam Hamidieh

  13. In Class Exercise 1 State the population parameter of interest and the appropriate H0 and Ha: A pharmaceutical company has recently developed a new drug for migraine headaches. The company claims that the mean time needed for the drug to enter the blood stream is less than 10 minutes. To convince the outside world, the company will conduct an experiment on a randomly chosen set of migraine headache sufferers. Faster absorption times are considered good. BUAD 310 - Kam Hamidieh

  14. In Class Exercise 2 State the population parameter interest and the appropriate H0 and Ha. A leasing firm operates on the industry assumption that the annual number of miles driven for leased cars is 14,000 miles on average. However, the firm has begun to doubt this assumption. The company likes to see if this assumption is still valid. The company will look at its own data to decide. BUAD 310 - Kam Hamidieh

  15. In Class Exercise 3 State the population parameter interest and the appropriate H0 and Ha. The mean weight of salmon grown at a commercial hatchery are generally around 7.5 pounds. However, this year the hatchery owner thinks that the mean weight is greater than 7.5 pounds. The hatchery owner will be choosing a random sample of fish to determine whether there is evidence to support their claim. BUAD 310 - Kam Hamidieh

  16. Trash Bags Example Continued • A sample of 40 new trash bags is randomly selected for which the mean breaking strength is calculated to be = 50.45. The sample SD is s = 1.65. The units are in pounds. H0: μ ≤ 50 vs. Ha: μ > 50 • Compute the test statistic: BUAD 310 - Kam Hamidieh

  17. Test Statistics • The test statistic summarize all of our information into one single number. • We can now make probability statements based on the test statistic. WHY? • It is a random variable! • Assuming the null hypothesis is true then: BUAD 310 - Kam Hamidieh

  18. P-Value P-Value = The probability of observing the value of the test statistic or values more extremes (which support the alternative hypothesis) assuming that the null hypothesis is true. Here T ~ t-distribution with df = n - 1 BUAD 310 - Kam Hamidieh

  19. Trash Bag Example Continued • For this example, when assuming null is true:T ~ t-distribution with df = 40 – 1 = 39 • Go to StatCrunch or use online resource:http://www.stat.tamu.edu/~west/applets/tdemo.html • Here: P(T > 1.725) ≈ 0.046 • Meaning: Assuming that the null hypothesis is true, the probability of getting the observed value of 1.725 or value more extreme that support the alternative is 0.046. • Is this compatible with our assumption that the null hypothesis is true: H0: μ ≤ 50 ? BUAD 310 - Kam Hamidieh

  20. Cut off or Tolerance Level • We will denote the cut off by α (alpha): α = P(Reject Null when Null is true) = P(Type I Error) • This cut off is generally set at a small values such as 5% or 1%. WHY? • Our decision: • If P-Value ≤ α, then the evidence provided by the data is against the null hypothesis. We will reject the null hypothesis. Stat jargon: we have a statistically significant result at level α. • If P-Value > α, then we fail to reject the null hypothesis. Our data does not provide enough evidence to reject the null. We do not have a statistically significant result at level α. BUAD 310 - Kam Hamidieh

  21. More on Trash • In the trash bag example, if we had set α = 5%, then:(P-value = 0.046) < (α= 0.05)Reject the null hypothesis. We have a statistically significant result at level 5%. • In summary: We have sufficient evidence to support the hypothesis that the mean strength of the new trash bags is greater than 50 pounds. (We’ve got better trash bags!) BUAD 310 - Kam Hamidieh

  22. Important Issues • You must decide on α level before seeing the data! WHY? • Should this statement make you squirm?“We proved that the mean strength of our new trash bags is greater than 50.” BUAD 310 - Kam Hamidieh

  23. Important Issues • Suppose you had gotten a P-Value of 0.15. Now you could not reject the null at level 5%. Could this make you squirm?“We accept that the mean strength of our new trash bags is still less than 50.” BUAD 310 - Kam Hamidieh

  24. Important Issues Continued • You should squirm severely when you hear this: “The p-value tells us the probability that the null hypothesis is true.” WHY? WRONG!!!! BUAD 310 - Kam Hamidieh

  25. Important Issues Continued • Note: everything we have done has hinged on: • We have a random sample. • In theory, data should come from normal population but t-tests are generally robust when n is large. • For the “largeness” of n refer to slide 25, lecture 13. • Could you suggest an alternative way to determine whether the new trash bags have improved (strength of over 50 pounds)? BUAD 310 - Kam Hamidieh

  26. In Summary • State the null hypothesis H0 and the alternative hypothesis Ha. The test is designed to assess the strength of the evidence against H0 in favor of Ha. • Calculate the value of the test statistic. • Assuming that the null hypothesis is true, find the probability that the test statistic is at least as extreme as the value you just calculated. This probability is called the P-value. • Compare the P-value to a fixed significance level α. If the P-value is less than α, then you reject H0 at significance level α. Otherwise, you do not have sufficient evidence to reject H0 at level α. BUAD 310 - Kam Hamidieh

  27. Example The credit manager of a large department store claims that the mean balance for the store’s charge account customers is $410. An independent auditor selects a random sample of 18accounts and finds a mean balance of $511 and a standard deviation of $184. If these data provide strong evidence against the manager’s claim, the auditor intends to examine all charge account balances. What action should the auditor take? Set α = 5%. BUAD 310 - Kam Hamidieh

  28. In Class Exercise 4 In a study entitled How Undergraduate Students Use Credit cards, it was reported that undergraduate have a mean credit card balance of $3170 (Sallie Mae, April 2009). A study has been conducted to determine if it can be concluded that the mean credit card balance for undergraduate students has increased compared to the April 2009. The results of the study are: = 3320, s = 1000, n = 180. Set α = 5%. • Is this (statistical) evidence that the mean credit card balance for the undergrads has increased compared to 2009? • In addition to the hypothesis test, create a 95% confidence interval for the current mean credit card balance for the undergrads. • Do your results from parts 1 and 2 agree? BUAD 310 - Kam Hamidieh

  29. Next Time • We will finish up Chapter 16. BUAD 310 - Kam Hamidieh

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