The t Test. Lecture 35 Section Robb T. Koether. Hampden-Sydney College. Mon, Oct 31, 2011
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1 The t Test Lecture 35 Section 10.2 Robb T. Koether Hampden-Sydney College Mon, Oct 31, 2011 Robb T. Koether (Hampden-Sydney College) The t Test Mon, Oct 31, / 38
2 Outline 1 Introduction 2 Hypothesis Testing (σ Unkwn) 3 The Decision Tree 4 The t-test on the TI-83 5 Testing for Normality 6 Assignment Robb T. Koether (Hampden-Sydney College) The t Test Mon, Oct 31, / 38
3 Outline 1 Introduction 2 Hypothesis Testing (σ Unkwn) 3 The Decision Tree 4 The t-test on the TI-83 5 Testing for Normality 6 Assignment Robb T. Koether (Hampden-Sydney College) The t Test Mon, Oct 31, / 38
4 Outline 1 Introduction 2 Hypothesis Testing (σ Unkwn) 3 The Decision Tree 4 The t-test on the TI-83 5 Testing for Normality 6 Assignment Robb T. Koether (Hampden-Sydney College) The t Test Mon, Oct 31, / 38
5 Hypothesis Testing (σ Unkwn) Example (Hypothesis Testing (σ Unkwn)) Test the hypotheses concerning the average AP Statistics score, assuming that the value of σ is t kwn. Assume that the population of test scores is rmal. Score Frequency Robb T. Koether (Hampden-Sydney College) The t Test Mon, Oct 31, / 38
6 Hypothesis Testing (σ Unkwn) Example (Hypothesis Testing (σ Unkwn)) (1) µ = average AP Statistics score. H 0 : µ = 3 H 1 : µ > 3 Robb T. Koether (Hampden-Sydney College) The t Test Mon, Oct 31, / 38
7 Hypothesis Testing (σ Unkwn) Example (Hypothesis Testing (σ Unkwn)) (1) µ = average AP Statistics score. H 0 : µ = 3 H 1 : µ > 3 (2) α = Robb T. Koether (Hampden-Sydney College) The t Test Mon, Oct 31, / 38
8 Hypothesis Testing (σ Unkwn) Example (Hypothesis Testing (σ Unkwn)) (1) µ = average AP Statistics score. H 0 : µ = 3 H 1 : µ > 3 (2) α = (3) t = x µ 0 s/ n. Robb T. Koether (Hampden-Sydney College) The t Test Mon, Oct 31, / 38
9 Hypothesis Testing (σ Unkwn) Example (Hypothesis Testing (σ Unkwn)) Use the TI-83 to compute the statistics x and s. We get x = s = Robb T. Koether (Hampden-Sydney College) The t Test Mon, Oct 31, / 38
10 Hypothesis Testing (σ Unkwn) Example (Hypothesis Testing (σ Unkwn)) (4) t = / 36 = = Robb T. Koether (Hampden-Sydney College) The t Test Mon, Oct 31, / 38
11 Hypothesis Testing (σ Unkwn) Example (Hypothesis Testing (σ Unkwn)) (4) t = / 36 = = (5) p-value = tcdf(1.0625,e99,35) = Robb T. Koether (Hampden-Sydney College) The t Test Mon, Oct 31, / 38
12 Hypothesis Testing (σ Unkwn) Example (Hypothesis Testing (σ Unkwn)) (4) t = / 36 = = (5) p-value = tcdf(1.0625,e99,35) = (6) Accept H 0. Robb T. Koether (Hampden-Sydney College) The t Test Mon, Oct 31, / 38
13 Hypothesis Testing (σ Unkwn) Example (Hypothesis Testing (σ Unkwn)) (4) t = / 36 = = (5) p-value = tcdf(1.0625,e99,35) = (6) Accept H 0. (7) The average AP Statistics score is t greater than 3. Robb T. Koether (Hampden-Sydney College) The t Test Mon, Oct 31, / 38
14 Hypothesis Testing (σ Unkwn) Example (Hypothesis Testing (σ Unkwn)) (4) t = / 36 = = (5) p-value = tcdf(1.0625,e99,35) = (6) Accept H 0. (7) The average AP Statistics score is t greater than 3. Had we used z instead of t, the p-value would have been Robb T. Koether (Hampden-Sydney College) The t Test Mon, Oct 31, / 38
15 Outline 1 Introduction 2 Hypothesis Testing (σ Unkwn) 3 The Decision Tree 4 The t-test on the TI-83 5 Testing for Normality 6 Assignment Robb T. Koether (Hampden-Sydney College) The t Test Mon, Oct 31, / 38
16 Three Tests We w have three different tests of hypotheses: Robb T. Koether (Hampden-Sydney College) The t Test Mon, Oct 31, / 38
17 Three Tests We w have three different tests of hypotheses: 1-sample z-test of proportions (1-PropZTest). Robb T. Koether (Hampden-Sydney College) The t Test Mon, Oct 31, / 38
18 Three Tests We w have three different tests of hypotheses: 1-sample z-test of proportions (1-PropZTest). 1-sample z-test of means (Z-Test). Robb T. Koether (Hampden-Sydney College) The t Test Mon, Oct 31, / 38
19 Three Tests We w have three different tests of hypotheses: 1-sample z-test of proportions (1-PropZTest). 1-sample z-test of means (Z-Test). 1-sample t-test of means (T-Test). Robb T. Koether (Hampden-Sydney College) The t Test Mon, Oct 31, / 38
20 Three Tests We w have three different tests of hypotheses: 1-sample z-test of proportions (1-PropZTest). 1-sample z-test of means (Z-Test). 1-sample t-test of means (T-Test). We need to be careful when deciding which test to use. Robb T. Koether (Hampden-Sydney College) The t Test Mon, Oct 31, / 38
21 Three Tests We w have three different tests of hypotheses: 1-sample z-test of proportions (1-PropZTest). 1-sample z-test of means (Z-Test). 1-sample t-test of means (T-Test). We need to be careful when deciding which test to use. It takes practice. Robb T. Koether (Hampden-Sydney College) The t Test Mon, Oct 31, / 38
22 The Decision Tree Is σ kwn? Is the population rmal? Come back later Z = σ / n Z σ / n Give up Robb T. Koether (Hampden-Sydney College) The t Test Mon, Oct 31, / 38
23 The Decision Tree Is σ kwn? Is the population rmal? Is the population rmal? Z = σ / n Z σ / n Give up Robb T. Koether (Hampden-Sydney College) The t Test Mon, Oct 31, / 38
24 The Decision Tree Is σ kwn? Is the population rmal? Is the population rmal? Z = σ / n Z σ / n Give up Robb T. Koether (Hampden-Sydney College) The t Test Mon, Oct 31, / 38
25 The Decision Tree Is σ kwn? Is the population rmal? Is the population rmal? Z = σ / n Z σ / n Give up T = s / n Z T Robb T. Koether (Hampden-Sydney College) The t Test Mon, Oct 31, / 38
26 The Decision Tree Is σ kwn? Is the population rmal? Is the population rmal? Z = σ / n Z σ / n Give up T = s / n Z T T = s / n Robb T. Koether (Hampden-Sydney College) The t Test Mon, Oct 31, / 38
27 The Decision Tree Is σ kwn? Is the population rmal? Is the population rmal? Z = σ / n Z σ / n Give up T = s / n Z T T = s / n Robb T. Koether (Hampden-Sydney College) The t Test Mon, Oct 31, / 38
28 The Decision Tree Is σ kwn? Is the population rmal? Is the population rmal? Z = σ / n Z σ / n Give up T = s / n Z T T = s / n Z s / n Robb T. Koether (Hampden-Sydney College) The t Test Mon, Oct 31, / 38
29 The Decision Tree Is σ kwn? Is the population rmal? Is the population rmal? Z = σ / n Z σ / n Give up T = s / n Z T T = s / n Z s / n Give up Robb T. Koether (Hampden-Sydney College) The t Test Mon, Oct 31, / 38
30 When to Use Z Which statistic? Use z when σ is kwn and the population is rmal ( matter how small the sample). Robb T. Koether (Hampden-Sydney College) The t Test Mon, Oct 31, / 38
31 When to Use Z Is σ kwn? Is the population rmal? Is the population rmal? Z = σ / n Z σ / n Give up T = s / n Z T T = s / n Z s / n Give up Robb T. Koether (Hampden-Sydney College) The t Test Mon, Oct 31, / 38
32 When to Use Z Which statistic? Use z when the population is t rmal, but the sample size is large (whether or t σ is kwn). Robb T. Koether (Hampden-Sydney College) The t Test Mon, Oct 31, / 38
33 When to Use Z Is σ kwn? Is the population rmal? Is the population rmal? Z = σ / n Z σ / n Give up T = s / n Z T T = s / n Z s / n Give up Robb T. Koether (Hampden-Sydney College) The t Test Mon, Oct 31, / 38
34 When to Use t Which statistic? Use t when σ is t kwn and the population is rmal ( matter how large or small the sample). If the sample size is small, then we must use t. If the sample size is large, then we may use z instead of t. Robb T. Koether (Hampden-Sydney College) The t Test Mon, Oct 31, / 38
35 When to Use t Is σ kwn? Is the population rmal? Is the population rmal? Z = σ / n Z σ / n Give up T = s / n Z T T = s / n Z s / n Give up Robb T. Koether (Hampden-Sydney College) The t Test Mon, Oct 31, / 38
36 When to Use t Is σ kwn? Is the population rmal? Is the population rmal? Z = σ / n Z σ / n Give up T = s / n Z T T = s / n Z s / n Give up Robb T. Koether (Hampden-Sydney College) The t Test Mon, Oct 31, / 38
37 When to Give up Which statistic? Give up only when the population is t rmal and the sample size is small. Robb T. Koether (Hampden-Sydney College) The t Test Mon, Oct 31, / 38
38 When to Give up Is σ kwn? Is the population rmal? Is the population rmal? Z = σ / n Z σ / n Give up T = s / n Z T T = s / n Z s / n Give up Robb T. Koether (Hampden-Sydney College) The t Test Mon, Oct 31, / 38
39 Summary Std. Dev. Population n Test σ Normal Any Z-Test σ Any 30 Z-Test s Normal < 30 T-Test s Normal 30 T-Test or Z-Test σ or s Not rmal < 30 Give up Robb T. Koether (Hampden-Sydney College) The t Test Mon, Oct 31, / 38
40 Outline 1 Introduction 2 Hypothesis Testing (σ Unkwn) 3 The Decision Tree 4 The t-test on the TI-83 5 Testing for Normality 6 Assignment Robb T. Koether (Hampden-Sydney College) The t Test Mon, Oct 31, / 38
41 TI-83 - Hypothesis Testing When σ is Unkwn TI-83 The t Test Press STAT. Select TESTS. Select T-Test. A window appears requesting information. Choose Data or Stats. Robb T. Koether (Hampden-Sydney College) The t Test Mon, Oct 31, / 38
42 TI-83 - Hypothesis Testing When σ is Unkwn TI-83 The t Test (Stats Option) Enter µ 0. Enter x. Enter s. (Remember, σ is unkwn.) Enter n. Select the alternative hypothesis and press ENTER. Select Calculate and press ENTER. Robb T. Koether (Hampden-Sydney College) The t Test Mon, Oct 31, / 38
43 TI-83 - Hypothesis Testing When σ is Unkwn TI-83 The t Test A window appears with the following information. The title T-Test The alternative hypothesis. The value of the test statistic t. The p-value. The sample mean. The sample standard deviation. The sample size. Robb T. Koether (Hampden-Sydney College) The t Test Mon, Oct 31, / 38
44 Example Example (The t-test) A sample of 18 students reveals the following AP Statistics scores. Score Frequency Furthermore, the value of σ is t kwn. The distribution in the sample suggests that the population is rmal. Test the hypothesis that the mean of all AP Statistics scores is greater than 3. Robb T. Koether (Hampden-Sydney College) The t Test Mon, Oct 31, / 38
45 Example (The t-test) A random sample of 10 hamburgers produced by a fast-food restaurant showed the following fat content, in grams: Assume that the fat content of all hamburgers from this restaurant has a rmal distribution. Test the hypothesis at the 1% level of significance that the average fat content of this restaurant s hamburgers is less than 32 grams. Show all seven steps. Robb T. Koether (Hampden-Sydney College) The t Test Mon, Oct 31, / 38
46 Outline 1 Introduction 2 Hypothesis Testing (σ Unkwn) 3 The Decision Tree 4 The t-test on the TI-83 5 Testing for Normality 6 Assignment Robb T. Koether (Hampden-Sydney College) The t Test Mon, Oct 31, / 38
47 Testing for Normality A researcher should t assume that the population is rmal unless he has some evidence. Given a small set of data, what would constitute evidence of rmality? Robb T. Koether (Hampden-Sydney College) The t Test Mon, Oct 31, / 38
48 Testing for Normality Definition (QQ Plot) A QQ plot is a graphical display of an actual data set versus an ideal data set of the same size from a rmal population. Numbers of equal rank are paired. If the QQ plot is very close to a straight line, then it is plausible that the data are from a rmal population. We will have to leave it to our own judgment whether the data are close eugh to a straight line. If the QQ plot indicates that we have rmality, then it longer matters whether the sample size is small. Robb T. Koether (Hampden-Sydney College) The t Test Mon, Oct 31, / 38
49 Example Example (QQ plots) A smaller sample of AP Statistics scores: Draw a QQ plot to test for rmality. If rmality holds, then test the claim at the 5% level that the population average score is greater than 3. Robb T. Koether (Hampden-Sydney College) The t Test Mon, Oct 31, / 38
50 Example Example (QQ plots) A small sample of AP Statistics scores is taken from a different school: Draw a QQ plot to test for rmality. If rmality holds, then test the claim at the 5% level that the population average score is greater than 3. Robb T. Koether (Hampden-Sydney College) The t Test Mon, Oct 31, / 38
51 Outline 1 Introduction 2 Hypothesis Testing (σ Unkwn) 3 The Decision Tree 4 The t-test on the TI-83 5 Testing for Normality 6 Assignment Robb T. Koether (Hampden-Sydney College) The t Test Mon, Oct 31, / 38
52 Assignment Homework Read Section 10.2, pages Let s Do It! 10.3, 10.4, Exercises 9-17, page 633. Robb T. Koether (Hampden-Sydney College) The t Test Mon, Oct 31, / 38
53 Assignment Answers 10. (1) µ is the average speed of drivers at this location. H 0 : µ = 70. H 1 : µ > 70. (2) α = (3) t = x µ 0 s/. [Note: The problem should have stated that the n population has a rmal distribution.] (4) t = (5) p-value = (6) Reject H 0. (7) The average speed of drivers that this location is greater than 70 mph. Robb T. Koether (Hampden-Sydney College) The t Test Mon, Oct 31, / 38
54 Assignment Answers 12. (a) H 0 : µ = 16 vs. H 1 : µ < 16. (b) It would be close to rmal if t for several larger values. (c) (1) H 0 : µ = 16. H 1 : µ < 16. (2) α = (3) t = x µ 0 s/ n. (4) t = (5) p-value = (6) Accept H 0. (7) The mean width percent for perch from this sea is 16%. Robb T. Koether (Hampden-Sydney College) The t Test Mon, Oct 31, / 38
55 Assignment Answers 14. (a) It is an outlier. (b) (1) H 0 : µ = 16. H 1 : µ < 16. (2) α = (3) t = x µ 0 s/ n. (4) t = (5) p-value = (6) Reject H 0. (7) The mean width percent for perch from this sea is less than 16%. (c) We rejected H 0 instead of accepted it. The sample mean decreased from to The sample standard deviation decreased from to The p-value decreased from to Robb T. Koether (Hampden-Sydney College) The t Test Mon, Oct 31, / 38
56 Assignment Answers 16. (a) H 0 : µ = 0. H 1 : µ < 0. (b) t = and p-value = (c) No. (d) The values between 4 and 4 are t far from a rmal distribution, but the values 10.1, 13.0, 19.4, and 42.7 distort the shape significantly. To use t, we need the assumption that the population is rmal, but that may t be the case. Robb T. Koether (Hampden-Sydney College) The t Test Mon, Oct 31, / 38
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