Hypothesis Testing (unknown σ)

Transcription

1 Hypothesis Testing (unknown σ) Business Statistics Recall: Plan for Today Null and Alternative Hypotheses Types of errors: type I, type II Types of correct decisions: type A, type B Level of Significance and Power of the Test Hypothesis testing with unknown σ Examples 1

2 Null and Alternative hypothesis Hypothesis : A statement about the value of a population parameter. In case of two hypotheses, the statement assumed to be true is called the null hypothesis (notation H0) and the contradictory statement is called the alternative hypothesis (notation Ha). Hypothesis testing : Based on sample evidence, a procedure for determining whether the hypothesis stated is a reasonable statement and should not be rejected, or is unreasonable and should be rejected. Null and Alternative Hypotheses The null hypothesis (H0) : It is a statement about the population that either is believed to be true or is used to put forth an argument unless it can be shown to be incorrect beyond a reasonable doubt. The alternative hypothesis (Ha): It is a claim about the population that is contradictory to H0 and what we conclude when we reject H0. 2

3 How to formulate the conclusion The conclusions are made in reference to the null hypothesis. We either reject the null hypothesis, or else we fail to reject the null hypothesis. If we reject the null hypothesis, it means that our data support the alternative hypotheses. If we fail to reject H0, it does not mean that it is proven to be true. It only means that our data fits within its scope. Errors and Correct Decisions Example: buying a laundry detergent. Two choices: Tide and generic detergent VS 3

4 Example: laundry detergents (continued) H0: there is no difference in performance Ha: Tide performs better H0 is true Type A correct decision. Decision: no difference in performance (and it is true). You bought generic product and saved money. Type I error (false positive). Decision: Tide performs better (in reality, no difference) You bought Tide and wasted $$ H0 is false Type II error (false negative). Decision: no difference (false, as Tide performs better). You bought generic product, got inferior results. Type B correct decision. Decision: Tide is better (true) You bought Tide and got better results Video on hypothesis testing with example: 4

5 Alpha and beta α is the level of significance of the test It equals the probability of making a type I error. 1 α equals the probability of making type A correct decision β is the probability of making a type II error. 1 β is called the Power of the Test. It equals the probability of making type B correct decision. Hypothesis testing when σ is not known When the population standard deviation, σ, is not known, we use the sample standard deviation, s, instead. And instead of the normal distribution, we use the Student s t distribution. For the classical approach, we will use Table 2 (Critical Values of Student s t Distribution), and for the p-value approach we will use Table 3 (called p-values for Student s t Distribution) with df = n 1 5

6 Hypothesis testing (classical approach) 1. State the hypotheses H0 and Ha. 2. Compute the test statistic t = x μ s n 3. Find the critical value(s) in the table. 4. Draw a bell-shaped curve and indicate the region(s) of rejection. 5. Place the test statistic onto the graph. 6. State your decision. Tails and critical values H a : μ μ 0 a two-tailed test with the critical values ±t( df, α 2) H a : μ > μ 0 a right-tailed test with the critical value t(df, α) H a : μ < μ 0 a left-tailed test with the critical value t(df, α) 6

7 Hypothesis testing (p-value approach) 1. State the hypotheses H0 and Ha. 2. Compute the test statistic t = x μ s n 3. Is this a left-, right-, or a two-tailed test? 4. Find the corresponding p-value in the table. 5. Compare the p-value with the level of significance α. 6. State your decision. What is the p-value? Right-tailed test: the area to the right of t Left-tailed test: the area to the left of t Two-tailed test: twice the area mathcaptain.com 7

8 Comparing the p-value and α: If the p-value is smaller than the level of significance α, it means that the test statistic is in the region of rejection. In this case, decision: reject H0. If the p-value is larger than the level of significance α, it means that the test statistic is in the region of acceptance. In this case, decision: fail to reject H0. Example: right-tailed test A company that makes Orange Soda claims that on the average its soda cans contain no more than 39 grams of sugar. A random sample of 12 cans tested in a lab yielded the following sugar contents (in grams): Can we disprove the company s claim at a 5% level of significance? Assume that the amount of sugar in cans is normally distributed. 8

9 Example: Orange Soda (classical approach) 1. H 0 : μ 39 g H a : μ > 39 g 2. Compute from sample: x = g, s = g Test statistic: t = / 12 = Since α = 0.05, we have the critical value t df, α = t 11, 0.05 = Plot the curve! 5. Where is t? (in the region of rejection) 6. Decision: reject H0 Example: Orange Soda (the p-value approach) 1. H 0 : μ 39 g H a : μ > 39 g 2. Test statistic t = This is a right-tailed test. 4. From Table 3 with df = 11: p-value = (always pick the bigger value) 5. Compare: p-value = < 0.05 = α 6. Decision: Reject H0 9

10 Example: left-tailed test A restaurant owner has counted the number of clients in his restaurant during 14 randomly selected Fridays: He wants to test the claim that the average number of clients in the restaurant on Fridays is actually smaller than 186, using a 2.5% level of significance. Assume that the number of clients is normally distributed. Example: restaurant (classical approach) 1. H 0 : μ 186 H a : μ < Compute from the sample: x = and s = Test statistic: t = = α = 0.025, the critical value: t df, α = t 13, = Plot the curve 5. Where is t? (in the region of acceptance) 6. Decision: Fail to reject H0 10

11 Example: restaurant (the p-value approach) 1. H 0 : μ 186 H a : μ < Test statistic: t = This is a left-tailed test. 4. Use Table 3 to find with df = 13 p-value = (take the larger value) 5. Compare: p-value = > = α 6. Decision: Fail to reject H0 Example: two-tailed test According to a website, the average weekly wage of government workers is $1038. A sample of randomly chosen 11 government workers is taken and their weekly wages are recorded: $1080 $990 $1540 $960 $920 $880 $1220 $1630 $940 $1270 $1160 Test at a 10% level of significance whether the average weekly wage of government workers is actually different from $1038. Assume a normal distribution of wages. 11

12 Example: wages (classical approach) 1. H 0 : μ = $1038 H a : μ $ Compute from the sample: x = $ and s = $252.6 Test statistic: t = = α = 0.1, the critical values: ±t df, α/2 = ±t 10, 0.05 = ± Plot the curve 5. Where is t? (in the region of acceptance) 6. Decision: Fail to reject H0 Example: wages (the p-value approach) 1. H 0 : μ = $1038 H a : μ $ Test statistic: t = This is a two-tailed test with positive t. 4. Find the p-value from Table 3 with df = 10: p-value = 2 * = (because two-tailed!) 5. Compare: p-value = > 0.1 = α 6. Decision: Fail to reject H0 The decision would be different if we would ve forgotten to multiply by 2. 12

13 Example: practice An article in the San Jose Mercury News claims that students in the California state university system take an average of 4.5 years to finish their undergraduate degrees. Suppose you suspect that the average time is longer. You conduct a survey of 30 students and obtain a sample mean of 4.8 years and a sample standard deviation of 0.9 years. Test the claim at a 5% level of significance. (Do both, the classical, and the p-value approaches.) Example: practice A website claims that in Toronto, the average water/garbage bill for a single family house is $650 per year. A researcher wants to test this claim by randomly choosing 50 single family houses. The average water/garbage bill in his sample turns out to be equal to $705 per year with a standard deviation of $195. Can he claim that the average annual water/garbage bill for a single family house in Toronto is actually different from $650 at a 2% level of significance? 13

14 Example: practice A telephone company claims that the mean duration of all long-distance phone calls made by its residential customers is at least 11 minutes. Here is a random sample of 16 long-distance calls, in minutes: Assuming the duration of long-distance calls is normally distributed, test at a 10% level of significance whether the mean duration of all long-distance calls is actually less than 11 minutes. 14