Chapter 3 Hypothesis Testing PDF

Business Statistics CHAPTER 3 HYPOTHESIS TESTING 1  What is Hypothesis? A hypothesis is an assumption about the population parameter. A parameter is a characteristic of the population, like its mean, proportion or variance. The parameter must be identified before...

Business Statistics CHAPTER 3 HYPOTHESIS TESTING 1  What is Hypothesis? A hypothesis is an assumption about the population parameter. A parameter is a characteristic of the population, like its mean, proportion or variance. The parameter must be identified before analysis. In business statistics, a hypothesis is a formal statement about a population parameter that we want to test. 2  Hypothesis Testing A hypothesis is an assumption about the population parameter (say population mean) which is to be tested. For that we collect sample data, then we calculate sample statistics (say sample mean) and then use this information to judge/decide whether hypothesized value of population parameter is correct or not. Hypothesis testing is a statistical method used to determine the likelihood of a hypothesis being true. In business statistics, it's a crucial tool for making data- driven decisions. 3 … Cont’d 4 … Cont’d 5  Null Hypothesis, Ho Null hypothesis is the default assumption or claim about a population parameter (mean, proportion, etc.) that is tested for possible rejection. Business decisions are often based on rejecting or failing to reject H0. State the Assumption (numerical) to be tested. E.g., The average weight of the semester 2 student is 58kgs (H₀: μ = 58) Begin with the assumption that the null hypothesis is TRUE. (Similar to the notion of innocent until proven guilty) 6  Alternative Hypothesis (H1) It is the opposite of null hypothesis. e.g. The average weight of the students is not equal to 58kgs. (H₁: μ ≠ 58) The alternative hypothesis contradicts H0 and represents what the researcher or analyst is trying to prove. It assumes a change, an effect, or a difference exists. 7 Steps of Hypothesis Testing 8 … Cont’d 9 Type I and Type II Errors 10  Examples of Type I & II Errors Type I Error (False Positive) A Type I error occurs when we reject a true null hypothesis. In simpler terms, it's like a false alarm. Example: Null Hypothesis (H₀): A new drug is ineffective. Alternative Hypothesis (H₁): The new drug is effective. If we conclude that the drug is effective (reject H₀) when it's actually not, we have made a Type I error. 11  … Cont’d Type II Error (False Negative) A Type II error occurs when we fail to reject a false null hypothesis. It's like missing a real effect. Example: Null Hypothesis (H₀): A new marketing campaign will not increase sales. Alternative Hypothesis (H₁): The new marketing campaign will increase sales. If we conclude that the campaign did not increase sales (fail to reject H₀) when it actually did, we've made a Type II error. 12 Summary of Type I & II Errors 13 One-Tailed & Two-Tailed Tests 14 … Cont’d 15 16 Z value for 1 tailed test & 2 tail test 17  Examples of Formulating Ho and Ha Example 1: The manager of a hotel has stated that the mean guest bill for a weekend is Birr 400 or less. A member of the hotel’s accounting staff has noticed that the total charges for guest bills have been increasing in recent months. The accountant will use a sample of weekend guest bills to test the manager’s claim. Required: State the null and alternative hypotheses: SOLUTION H₀:The mean guest bill for a weekend is Birr 400 or less. / H₀ : μ ≤ 400/ H1:The mean guest bill for a weekend is greater than Birr 400. / H1 : μ > 400/ 18  (Contd …) Example 2: Production workers at XY Company have been trained in their jobs by using two different training programs. The company training director would like to know whether there is a difference in mean productivity for workers trained in the two programs. Required: Develop the Ho and Ha SOLUTION H₀: There is no difference in the mean productivity of workers trained in the two programs. /Ho​: μ1​=μ2​/ H1: There is a difference in the mean productivity of workers trained in the two programs. /H1: μ1≠μ2​/ 19  Contd… Example 3: The manager at a drugstore assumes that the company’s employees are honest. However, there have been many shortages from the cash register lately. Required: Specify the null and alternative hypothesis SOLUTION H₀: The company’s employees are honest. /H₀:No shortages are due to dishonesty. / H1: The company’s employees are not honest. /H1: Shortages are due to dishonesty/ 20  HYPOTHESIS TESTING about : Population Normal,  known, n Small Example 1: Matador-Addis Tire Share Company claims that its tires have a mean life of 35,000 miles. A random sample of 16 of these tires is tested if the sample mean in 33,000 miles. Assume that the population standard deviation is 3000 miles and the lives of tires are approximately normally distributed. Test the share company’s claim using a 5% level of significance. 21 Solutions to Example 1 22  … cont’d Example 2: A Teachers’ union is on strike for higher wages. The union claims that the mean salary for teachers is at most Birr 8,400 per year. The legislator does not want to reject the union’s claim, however, unless the evidence is very strong against if. Assume that salaries follow a normal distribution and the population standard deviation is known to be Birr 3000. A random sample of 64 teachers is obtained, and the sample mean is Birr, 9,400. Test if the state legislator accepts the unions’ claim or not at 1% significance level. 23 Solutions to Example 2 24  contd… Example 3:A fertilizer company claims that the use of its product will result in a yield of at least 35 quintals of wheat per hectare, on average, Application at the fertilizer to a randomly selected 36 sample hectares resulted in a yield of 34quintals per hectare. Assume the population standard deviation is 5 quintals and yields per hectare are normally distributed. Test the company’s claim at 1% level of significance. 25 Solutions to Example 3 26  HYPOTHESIS TESTING about : Population Normal,  Unknown, n Small Example1: A contractor assumes that construction workers are idle for 75 minutes or less per day. A random sample of 25 construction workers was taken and the mean idle time was found to be 84 minutes per day with a sample standard deviation of 20 minutes. Assume that the population is approximately normally distributed; use a 5% level of significance to test the contractor’s assumption. 27 Solution to example 1 28  HYPOTHESIS TESTING about : Population Normal,  Unknown, N Small Example 2: A director of a secretarial school Assumes that its graduates can type at least 50 words per minute on average. Suppose you want to hire some of these graduates if the director’s claim is true; and you test the typing speed of 18 of the graduates and obtain a mean of 40 wards per minute with a sample variance of 720. Assuming the typing speed for the graduates of the secretarial school is normally distributed, test the director’s claim and decide whether to hire the graduates or not, using a 5% level of significance. 29 Solution to example 2 30 Example3:A manager of star supermarket assumes that the trip of shoppers takes a mean of 22 minutes. Suppose that in an effort to test this claim, a sample of 27 local shoppers are taken and their mean shopping trip was 30 minutes with sample standard deviation 10 minutes. Use a 2% level of significance and test the claim whether there is difference in the mean shopping trips. 31 Solution to example 3 32 33  Hypothesis testing about a population proportion (p). Similar to that of hypothesis testing about a population mean, hypothesis testing about a population proportion has three terms. 1. Ho: P = y 2. Ho: P  y 3. Ho: P  y Ha: P  y Ha: Py Ha: P y 34  Example 1: 1. A magazine claims that 25% of its readers are college students. A random sample of 200 readers is taken. It is found that 42 of these readers are college students. Use a 10% level of significance and test the magazine’s claim. 35 Solution to Example 1 36 37 Example 2: An Economist states that more than 35% of Addis’s labor force in unemployed. You don’t know if the economist’s estimate is too high or too low. Thus, you want to test the economist’s claim using a 5% level of significance. You obtain a random sample of 400 people in the labor force, of whom 128 are unemployed. Would you reject the economist’s claim? 38 Solution to Example 2 39 40 Example 3. A survey of the morning beverage market has shown that the primary breakfast beverage for 60% of Ethiopian town and city dwellers is tea. Ethiopian coffee and Tea Authority believes that the figure is higher for Addis. To test this idea, one of the employees of Ethiopian coffee and Tea Authority contacts a random sample of 500 residents in Addis and asks which primary beverage they consumed for breakfast that day. Suppose 325 replied that tea was the primary beverage. Using a 0.01 level of significance, test the idea that the tea figure is higher for Addis. 41 Solution to Example 3 42 43 End of Chapter 3 Thanks!