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MKT 504
Roy Morley

QMS202-Business Statistics II Chapter11 Chapter11 Fundamentals of Hypothesis Testing: One-Sample tests Outcomes: 1. Define a hypothesis and hypothesis testing 2. Describe the hypothesis-testing procedure 3. Distinguish between a one-tailed and a two-tailed test of hypothesis 4. Conduct a hypothesis test for a population mean with known population standard deviation 5. Conduct a hypothesis test for a population mean with unknown population standard deviation 6. Conduct a hypothesis test for a population proportion 7. Define Type I and Type II errors 8. Compute and interpret a p-value What is a Hypothesis? 1. A hypothesis is a statement about a population. 2. In statistical analysis we make a claim, that is, state a hypothesis, collect data, and then use the data to test the claim. 3. In most cases the population is so large that it is not feasible to study all the items, objects or persons in the population. For example, it would not be possible to contact every accountant in Canada to find out his or her annual income. 4. An alternative to measuring or interviewing the entire population is to take a sample from the population and test a statement to determine whether the sample does or does not support the statement concerning the population. What is Hypothesis Testing? 1. A procedure based on sample evidence and probability theory which determine whether the hypothesis is a reasonable statement. 2. The procedure for testing a hypothesis Step1. Define the parameter(s) of interest Step2. State null and alternative hypotheses Step3. Identify a level of significance Step4. Identify the test statistic Step5. Compute the value of test statistic and p-value Step6. State the statistical decision and business conclusion Winter2013 page#1 QMS202-Business Statistics II Chapter11 Note: 1. Null Hypothesis H o: A statement about the value of a population parameter Alternative Hypothesis H : -The opposite of the null hypothesis A - The hypothesis that the researcher wishes to support. [The symbol for Alternative Hypothesis in the textbook is ] 1 The goal of the process is to determine whether there is enough evidence to infer that the alternative hypothesis is true 2. Level of significance (α): - When a null hypothesis is rejected, the test is said to be statistically significant at that significance level - To determine the rejection region. - Set by researcher before looking at the data 3. Test Statistics: A value, computed from sample information, used to determine whether or not to reject the null hypothesis 4. p-value : the probability of observing a test statistic at least as extreme as, or more extreme than, the one computed given that the null hypothesis is true [ i. If the p-value is less th, then reject ; o ii. If the p-value is more tha, then do not reject o] 5. There are TWO approaches for drawing a statistical decision i. Critical value approach ii. p-value approach 6. There are two possible business conclusions: i. Conclude that there is enough evidence to support the alternative hypothesis. ii. Conclude that there is not enough evidence to support the alternative hypothesis. 7. Type I error: Rejecting the null hypothesis when it is true 8. Type II error: Do not reject the null hypothesis when it is false Winter2013 page#2 QMS202-Business Statistics II Chapter11 Rejection region a. For one-tailed test, use α for calculating critical value α right-tailed test left-tailed test critical value critical value α a. For two-tailed test, use for calculating critical values 2 critical value critical value Risks in Decision Making Winter2013 page#3 QMS202-Business Statistics II Chapter11 When we use sample data to make decisions about population parameters, there is always a risk of reaching incorrect conclusions. Two types of errors can happen in hypothesis testing. Type I error 1.Occurs when we reject H 0, when it is true. 2.The probability of occurring a type I error is α . This is called the level of significance of the statistical test. 3.We control the type I error by deciding the value ofα that we want to have of rejectingH 0, when it is true. 4.Typically,α takes values such as 0.01, 0.05 or 0.1. α is the probability of rejectingH 0when it is true. Type II error 1.Occurs when we do not reject H 0when it is false. 2.The probability of a type II error .s 3.One way to reduce type II error is to increase the sample size. 4.Large samples generally permit to detect small differences between hypothesized values and actual parameters. The β risk The probability of committing a type II error i. If the difference between the hypothesized and the actual value of the population parameter is large, thenβ is small. Actual Situation Statistical Decision H 0True H 0False Do not reject H0 Correct decision Type II error Confidence = 1−α P(Type II error) = Reject H0 Type I error Correct decision P(Type I error) =α Power = (1−β ) Winter2013 page#4 QMS202-Business Statistics II Chapter11 Example1 The BigMacBurger restaurant chain claims that the waiting time of all customers for service is normally distributed, with a population mean of waiting time of 4 minutes and a population standard deviation of 1.4 minutes. The quality assurance department found in a sample of 60 customers at Yonge and Dundas that the mean waiting time was 4.1 minutes. Can we conclude that the mean waiting time is more than 4 minutes? Test at 5% of level of significance. Calculator Output 1-Sample z Test μ > 4 z = 0.55328 p = 0.29003 x = 4.1 n = 60 Step1 μ Let be the population mean of waiting time Step2 H o μ ≤ 4 H A μ > 4 Step3 Level of significance = 0.05 Step 4 one-sample mean z test Step5 zstat 0.55328 p-value = 0.29003 x − μ 4.1− 4 zstat σ = 1.4 = 0.55328 p-value = P (z > 0.55328)= 0.29003 n 60 zcritical.64485 Step6 Since the p-value > 0.05, do not reject the null hypothesis. There is not enough evidence to conclude that the population mean of waiting time is more than 4 minutes. Winter2013 page#5 QMS202-Business Statistics II Chapter11 Example2 A recent survey found that young professional working on Bay Street watched an average of 6.8 DVDs per month. A random sample of 36 young professionals revelaed that the mean number of DVDs watched last month was 6.2, with a population standard deviation of 2.5. Can we conclude that young professionals α on Bay Street watch fewer than 6.8 DVDs? Use =0.05 Calculator Output 1-Sample z Test μ < 6.8 z = -1.44 p = 0.074933 x = 6.2 n = 36 Step1 Let μ Step2 H o μ H A μ Step3 Level of significance = Step 4 Step5 zstat p-value = x − μ zstat σ = n zcritical Step6 Winter2013 page#6 QMS202-Business Statistics II Chapter11 T Test of Hypothesis for the Mean (population standard deviation σ unknown) Example3 The Ryerson’s Discount Appliance Store issues its own credit card. The credit card manager wants to find whether the mean monthly-unpaid balance is more than $500. The level of significance is set at 0.05. A random check of 180 unpaid balances revealed the sample mean is $508 and the standard deviation of the sample is $47. It is known that the monthly-unpaid balance is normally distributed. Should the credit card manager conclude that the population mean is greater than $500? Calculator Output 1-Sample t Test μ > 500 t = 2.2836 p = 0.011783 x = 508 sx = 47 n = 180 Step1 Let μ Step2 H o μ H : μ A Step3 Level of significance = Step 4 Step5 t = p-value = stat x − μ t s stat = n d.f = t = critical Step6 Winter2013 page#7 QMS202-Business Statistics II Chapter11 Example4 At the time John was hired as a server at the Ryerson Family restaurant at Dundas and Bay, he was told” You can earn average of not more than $120 a day in tips.” Over the first 40 days he was employed at the restaurant, the mean daily amount of his tips was $120.50 and a standard deviation of $41.32. It is known that the daily amount of tips is normally distributed. At the 0.05 significance level, can John conclude that he is earning an average of more than $120 in tips per day in the entire year? Calculator Output 1-Sample t Test μ > 120 t = 0.076531 p = 0.46969 x = 120.5 sx = 41.32 n = 40 Step1 Let μ Step2 H o μ H A μ Step3 Level of significance = Step 4 Step5 tstat p-value = d.f = t critical Step6 Winter2013 page#8 QMS202-Busines
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