# 5/31/2013. Chapter 8 Hypothesis Testing. Hypothesis Testing. Hypothesis Testing. Outline. Objectives. Objectives

Size: px
Start display at page:

Download "5/31/2013. Chapter 8 Hypothesis Testing. Hypothesis Testing. Hypothesis Testing. Outline. Objectives. Objectives"

Transcription

1 C H 8A P T E R Outline 8 1 Steps in Traditional Method 8 2 z Test for a Mean 8 3 t Test for a Mean 8 4 z Test for a Proportion 8 6 Confidence Intervals and Copyright 2013 The McGraw Hill Companies, Inc. Permission required for reproduction or display. 1 C H 8A P T E R Objectives 1 Understand the definitions used in hypothesis testing. 2 State the null and alternativehypotheses. 3 Find critical values for the z test. 4 State the five steps used in hypothesis testing. 5 Test means when is known, using the z test. 6 Test means when is unknown, using the t test. 7 Test proportions, using the z test. C H 8A P T E R Objectives 9 Test hypotheses, using confidence intervals. Researchers are interested in answering many types of questions. For example, Is the earth warming up? Does a new medication lower blood pressure? Does the public prefer a certain color in a new fashion line? Is a new teaching technique better than a traditional one? Do seat belts reduce the severity of injuries? These types of questions can be addressed through statistical hypothesis testing, which is a decision-making process for evaluating claims about a population. Three methods used to test hypotheses: 1. The traditional method 2. The P-value method 3. The confidence interval method Bluman, 5 Bluman, 6 1

2 8.1 Steps in - Traditional Method A statistical hypothesis is a conjecture about a population parameter. This conjecture may or may not be true. The null hypothesis, symbolized by H 0, is a statistical hypothesis that states that there is no difference between a parameter and a specific value, or that there is no difference between two parameters. The alternative hypothesis is symbolized by H 1. Steps in - Traditional Method The alternative hypothesis, symbolized by H 1, is a statistical hypothesis that states the existence of a difference between a parameter and a specific value, or states that there is a difference between two parameters. Bluman, 7 Bluman, 8 Situation A A medical researcher is interested in finding out whether a new medication will have any undesirable side effects. The researcher is particularly concerned with the pulse rate of the patients who take the medication. Will the pulse rate increase, decrease, or remain unchanged after a patient takes the medication? The researcher knows that the mean pulse rate for the population under study is 82 beats per minute. The hypotheses for this situation are H : 82 0 H : 82 1 This is called a two-tailedtailed hypothesis test. Situation B A chemist invents an additive to increase the life of an automobile battery. The mean lifetime of the automobile battery without the additive is 36 months. In this book, the null hypothesis is always stated using the equals sign. The hypotheses for this situation are H : 36 0 H : 36 1 This is called a right-tailedtailed hypothesis test. Bluman, 9 Bluman, 10 Situation C A contractor wishes to lower heating bills by using a special type of insulation in houses. If the average of the monthly heating bills is \$78, her hypotheses about heating costs with the use of insulation are H : 78 H : This is called a left-tailedtailed hypothesis test. Claim When a researcher conducts a study, he or she is generally looking for evidence to support a claim. Therefore, the claim should be stated as the alternative hypothesis, or research hypothesis. A claim, though, can be stated as either the null hypothesis or the alternative hypothesis; however, the statistical evidence can only support the claim if it is the alternative hypothesis. Statistical evidence can be used to reject the claim if the claim is the null hypothesis. These facts are important when you are stating the conclusion of a statistical study. Bluman, 11 Bluman, 12 2

3 After stating the hypotheses, the researcher s next step is to design the study. The researcher selects the correct statistical test, chooses an appropriate level of significance, and formulates a plan for conducting the study. A statistical test uses the data obtained from a sample to make a decision about whether the null hypothesis should be rejected. The numerical value obtained from a statistical test is called the test value. In the hypothesis-testing situation, there are four possible outcomes. Bluman, 13 Bluman, 14 In reality, the null hypothesis may or may not be true, and a decision is made to reject or not to reject it on the basis of the data obtained from a sample. A type I error occurs if one rejects the null hypothesis when it is true. A type II error occurs if one does not reject the null hypothesis when it is false. Bluman, 15 Bluman, 16 The level of significance is the maximum probability of committing a type I error. This probability is symbolized by (alpha). That is, P(type I error) =. Likewise, P(type II error) = (beta). Typical significance levels are: 0.10, 0.05, and 0.01 For example, pe, when = 0.10, 0, there e is a 10% chance of rejecting a true null hypothesis. Bluman, 17 Bluman, 18 3

4 The critical value, C.V., separates the critical region from the noncritical region. The critical or rejection region is the range of values of the test value that indicates that there is a significant difference and that the null hypothesis should be rejected. The noncritical or nonrejection region is the range of values of the test value that indicates that the difference was probably due to chance and that the null hypothesis should not be rejected. Finding the Critical Value for α = 0.01 (Right-Tailed Test) z = 2.33 for α = 0.01 (Right-Tailed Test) Bluman, 19 Bluman, 20 Finding the Critical Value for α = 0.01 (Left-Tailed Test) Finding the Critical Value for α = 0.01 (Two-Tailed Test) z z = ±2.58 Because of symmetry, z = 2.33 for α = 0.01 (Left-Tailed Test) Bluman, 21 Bluman, 22 Procedure Table Finding the Critical Values for Specific α Values, Using Table E Step 1 Draw the figure and indicate the appropriate area. a. If the test is left tailed, the critical region, with an area equal to α, will be on the left side of the mean. b. If the test is right tailed, tailed the critical region, with an area equal to α, will be on the right side of the mean. c. If the test is two tailed, α must be divided by 2; onehalf of the area will be to the right of the mean, and one half will be to the left of the mean. Procedure Table Finding the Critical Values for Specific α Values, Using Table E Step 2 a. For a left tailed test, use the z value that corresponds to the area equivalent to α in Table E. b. For a right tailed test, use the z value that corresponds to the area equivalent to 1 α. c. For a two tailed test, use the z value that corresponds to α/2 for the left value. It will be negative. For the right value, use the z value that corresponds to the area equivalent to 1 α/2. It will be positive. 4

5 Section 8-1 Example 8-2 Page #408 Example 8-2: Using Table E Using Table E in Appendix C, find the critical value(s) for each situation and draw the appropriate figure, showing the critical region. a. A left-tailed test with α = Step 1 Draw the figure and indicate the appropriate area. Step 2 Find the area closest to in Table E. In this case, it is The z value is Bluman, 25 Bluman, 26 Example 8-2: Using Table E Using Table E in Appendix C, find the critical value(s) for each situation and draw the appropriate figure, showing the critical region. b. A two-tailed test with α = Step 1 Draw the figure with areas α /2 = 0.02/2 = Example 8-2: Using Table E Using Table E in Appendix C, find the critical value(s) for each situation and draw the appropriate figure, showing the critical region. c. A right-tailed test with α = Step 1 Draw the figure and indicate the appropriate area. Step 2 Find the areas closest to 0.01 and The areas are and The z values are 2.33 and Step 2 Find the area closest to 1 α = There is a tie: and Average the z values of 2.57 and 2.58 to get or Bluman, 27 Bluman, 28 Procedure Table Solving Hypothesis Testing Problems (Traditional Method) Step 1 State the hypotheses and identify the claim. Step 2 Find the critical value(s) from the appropriate table in Appendix C. Step 3 Compute the test value. Step 4 Step 5 Make the decision to reject or not reject the null hypothesis. Summarize the results. 8.2 z Test for a Mean The z test is a statistical test for the mean of a population. It can be used when n 30, or when the population is normally distributed and is known. The formula for the z test is X z n where X = sample mean μ = hypothesized population mean = population standard deviation n = sample size Bluman, 30 5

6 Section 8-2 Example 8-3 Page #412 Example 8-3: Days on Dealers Lots A researcher wishes to see if the mean number of days that a basic, low-price, small automobile sits on a dealer s lot is 29. A sample of 30 automobile dealers has a mean of 30.1 days for basic, low-price, small automobiles. At α = , test the claim that the mean time is greater than 29 days. The standard deviation of the population is 3.8 days. Bluman, 31 Bluman, 32 Example 8-3: Days on Dealers Lots Step 1 State the hypotheses and identify the claim. Step 2 Find the critical value. Since α = 0.05 and the test is a right-tailed test, the critical value is z= Example 8-3: Days on Dealers Lots Step 4 Make the decision. Since the test value, +1.59, is less than the critical value, +1.65, and is not in the critical region, the decision is to not reject the null hypothesis. Step 3 Compute the test value. Bluman, 33 Bluman, 34 Example 8-3: Days on Dealers Lots Step 5 Summarize the results. There is not enough evidence to support the claim that the mean time is greater than 29 days. Important Comments Even though in Example 8 3 the sample mean of 30.1 is higher than the hypothesized population mean of 29, it is not significantly higher. Hence, the difference may be due to chance. When the null hypothesis is not rejected, there is still a probability of a type II error, i.e., of not rejecting the null hypothesis when it is false. Bluman, 35 Bluman, 36 6

7 Section 8-2 Example 8-4 Page #413 Example 8-4: Cost of Men s Shoes A researcher claims that the average cost of men s athletic shoes is less than \$80. He selects a random sample of 36 pairs of shoes from a catalog and finds the following costs (in dollars). (The costs have been rounded to the nearest dollar.) Is there enough evidence to support the researcher s claim at α = 0.10? Assume = Step 1: State the hypotheses and identify the claim. H 0 : μ = \$80 and H 1 : μ < \$80 (claim) Bluman, 37 Bluman, 38 Example 8-4: Cost of Men s Shoes A researcher claims that the average cost of men s athletic shoes is less than \$80. He selects a random sample of 36 pairs of shoes from a catalog and finds the following costs (in dollars). (The costs have been rounded to the nearest dollar.) Is there enough evidence to support the researcher s claim at α = 0.10? Assume = Step 2: Find the critical value. Since α = 0.10 and the test is a left-tailed test, the critical value is z = Example 8-4: Cost of Men s Shoes A researcher claims that the average cost of men s athletic shoes is less than \$80. He selects a random sample of 36 pairs of shoes from a catalog and finds the following costs (in dollars). (The costs have been rounded to the nearest dollar.) Is there enough evidence to support the researcher s claim at α = 0.10? Assume = Step 3: Compute the test value. Using technology, we find X = 75.0 and = z X n Bluman, 39 Bluman, 40 Example 8-4: Cost of Men s Shoes Since the test value, 1.56, falls in the critical region, the decision is to reject the null hypothesis. There is enough evidence to support the claim that the average cost of men s athletic shoes is less than \$80. Section 8-2 Example 8-5 Page #414 Bluman, 41 Bluman, 42 7

8 Example 8-5: Cost of Rehabilitation The Medical Rehabilitation Education Foundation reports that the average cost of rehabilitation for stroke victims is \$24,672. To see if the average cost of rehabilitation is different at a particular hospital, a researcher selects a random sample of 35 stroke victims at the hospital and finds that the average cost of their rehabilitation is \$26,343. The standard deviation of the population is \$3251. At α = 0.01, can it be concluded that the average cost of stroke rehabilitation at a particular hospital is different from \$24,672? Step 1: State the hypotheses and identify the claim. H 0 : μ = \$24,672 and H 1 : μ \$24,672 (claim) Bluman, 43 Example 8-5: Cost of Rehabilitation The Medical Rehabilitation Education Foundation reports that the average cost of rehabilitation for stroke victims is \$24,672. To see if the average cost of rehabilitation is different at a particular hospital, a researcher selects a random sample of 35 stroke victims at the hospital and finds that the average cost of their rehabilitation is \$26,343. The standard deviation of the population is \$3251. At α = 0.01, can it be concluded that the average cost of stroke rehabilitation at a particular hospital is different from \$24,672? Step 2: Find the critical value. Since α = 0.01 and a two-tailed test, the critical values are z = ±2.58. Bluman, 44 Example 8-5: Cost of Rehabilitation Step 3: Find the test value. z X 26,343 24, n Example 8-5: Cost of Rehabilitation Reject the null hypothesis, since the test value falls in the critical region. There is enough evidence to support the claim that the average cost of rehabilitation at the particular hospital is different from \$24,672. Bluman, 45 Bluman, 46 The P-value (or probability value) is the probability of getting a sample statistic (such as the mean) or a more extreme sample statistic in the direction of the alternative hypothesis when the null hypothesis is true. In this section, the traditional method for solving hypothesis-testing problems compares z-values: critical value test value Test Value P-Value The P-value method for solving hypothesistesting problems compares areas: alpha P-value Bluman, 47 Bluman, 48 8

9 Procedure Table Solving Hypothesis Testing Problems (P Value Method) Step 1 State the hypotheses and identify the claim. Step 2 Compute the test value. Step 3 Step 4 Step 5 Find the P value. Make the decision. Summarize the results. Section 8-2 Example 8-6 Page #417 Bluman, 50 Example 8-6: Cost of College Tuition A researcher wishes to test the claim that the average cost of tuition and fees at a four-year public college is greater than \$5700. She selects a random sample of 36 four-year public colleges and finds the mean to be \$5950. The population standard deviation is \$659. Is there evidence to support the claim at a 0.05? Use the P-value method. Step 1: State the hypotheses and identify the claim. H 0 : μ = \$5700 and H 1 : μ > \$5700 (claim) Example 8-6: Cost of College Tuition A researcher wishes to test the claim that the average cost of tuition and fees at a four-year public college is greater than \$5700. She selects a random sample of 36 four-year public colleges and finds the mean to be \$5950. The population standard deviation is \$659. Is there evidence to support the claim at a 0.05? Use the P-value method. Step 2: Compute the test value. z X n Bluman, 51 Bluman, 52 Example 8-6: Cost of College Tuition A researcher wishes to test the claim that the average cost of tuition and fees at a four-year public college is greater than \$5700. She selects a random sample of 36 four-year public colleges and finds the mean to be \$5950. The population standard deviation is \$659. Is there evidence to support the claim at a 0.05? Use the P-value method. Step 3: Find the P-value. Using Table E, find the area for z = The area is Subtract from to find the area of the tail. Hence, the P-value is = Bluman, 53 Example 8-6: Cost of College Tuition Since the P-value is less than 0.05, the decision is to reject the null hypothesis. There is enough evidence to support the claim that the tuition and fees at four-year public colleges are greater than \$5700. Note: If α = 0.01, the null hypothesis would not be rejected. Bluman, 54 9

10 Section 8-2 Example 8-7 Page #418 Example 8-7: Wind Speed A researcher claims that the average wind speed in a certain city is 8 miles per hour. A sample of 32 days has an average wind speed of 8.2 miles per hour. The standard deviation of the population is 0.6 mile per hour. At α = 0.05, is there enough evidence to reject the claim? Use the P-value method. Step 1: State the hypotheses and identify the claim. H 0 : μ = 8 (claim) and H 1 : μ 8 Step 2: Compute the test value. z X n Bluman, 55 Bluman, 56 Example 8-7: Wind Speed A researcher claims that the average wind speed in a certain city is 8 miles per hour. A sample of 32 days has an average wind speed of 8.2 miles per hour. The standard deviation of the population is 0.6 mile per hour. At α = 0.05, is there enough evidence to reject the claim? Use the P-value method. Step 3: Find the P-value. The area for z = 1.89 is Subtract: = Since this is a two-tailed test, the area of must be doubled to get the P-value. The P-value is 2(0.0294) = Example 8-7: Wind Speed The decision is to not reject the null hypothesis, since the P-value is greater than There is not enough evidence to reject the claim that the average wind speed is 8 miles per hour. Bluman, 57 Bluman, 58 Guidelines for P-Values With No α If P-value 0.01, reject the null hypothesis. The difference is highly significant. If P-value > 0.01 but P-value 0.05, reject the null hypothesis. The difference is significant. If P-value > 0.05 but P-value 0.10, consider the consequences of type I error before rejecting the null hypothesis. If P-value > 0.10, do not reject the null hypothesis. The difference is not significant. Significance The researcher should distinguish between statistical significance and practical significance. When the null hypothesis is rejected at a specific significance level, it can be concluded that the difference is probably not due to chance and thus is statistically significant. However, the results may not have any practical significance. It is up to the researcher to use common sense when interpreting the results of a statistical test. Bluman, 59 Bluman, 60 10

11 8.3 t Test for a Mean The t test is a statistical test for the mean of a population and is used when the population is normally or approximately normally distributed, is unknown. The formula for the t test is X t s n The degrees of freedom are d.f. = n 1. Note: When the degrees of freedom are above 30, some textbooks will tell you to use the nearest table value; however, in this textbook, you should round down to the nearest table value. This is a conservative approach. Section 8-3 Example 8-8 Page #426 Bluman, 61 Bluman, 62 Example 8-8: Table F Find the critical t value for α = 0.05 with d.f. = 16 for a right-tailed t test. Find the 0.05 column in the top row and 16 in the left-hand column. The critical t value is Section 8-3 Examples 8-9 & 8-10 Page #426 Bluman, 63 Bluman, 64 Example 8-9: Table F Find the critical t value for α = 0.01 with d.f. = 22 for a lefttailed test. Find the 0.01 column in the One tail row, and 22 in the d.f. column. The critical value is t = since the test is a one-tailed left test. Example 8-10: Table F Find the critical value for α = 0.10 with d.f. = 18 for a twotailed t test. Find the 0.10 column in the Two tails row, and 18 in the d.f. column. The critical values are and Section 8-3 Example 8-12 Page #427 Bluman, 65 Bluman, 66 11

12 Example 8-12: Hospital Infections A medical investigation claims that the average number of infections per week at a hospital in southwestern Pennsylvania is A random sample of 10 weeks had a mean number of 17.7 infections. The sample standard deviation is 1.8. Is there enough evidence to reject the investigator s claim at α = 0.05? Step 1: State the hypotheses and identify the claim. H 0 : μ = 16.3 (claim) and H 1 : μ 16.3 Step 2: Find the critical value. The critical values are and for α = 0.05 and d.f. = 9. Example 8-12: Hospital Infections A medical investigation claims that the average number of infections per week at a hospital in southwestern Pennsylvania is A random sample of 10 weeks had a mean number of 17.7 infections. The sample standard deviation is 1.8. Is there enough evidence to reject the investigator s claim at α = 0.05? Step 3: Find the test value. z X s n Bluman, 67 Bluman, 68 Example 8-12: Hospital Infections Reject the null hypothesis since 2.46 > There is enough evidence to reject the claim that the average number of infections is Section 8-3 Example 8-13 Page #428 Bluman, 69 Bluman, 70 Example 8-13: Substitute Salaries An educator claims that the average salary of substitute teachers in school districts in Allegheny County, Pennsylvania, is less than \$60 per day. A random sample of eight school districts is selected, and the daily salaries (in dollars) are shown. Is there enough evidence to support the educator s claim at α = 0.10? Step 1: State the hypotheses and identify the claim. H 0 : μ = 60 and H 1 : μ < 60 (claim) Step 2: Find the critical value. At α = 0.10 and d.f. = 7, the critical value is Example 8-13: Substitute Salaries Step 3: Find the test value. Using the Stats feature of the TI-84, we find X = and s = t X s n Bluman, 71 Bluman, 72 12

13 Example 8-12: Substitute Salaries Do not reject the null hypothesis since falls in the noncritical region. There is not enough evidence to support the claim that the average salary of substitute teachers in Allegheny County is less than \$60 per day. Section 8-3 Example 8-16 Page #430 Bluman, 73 Bluman, 74 Example 8-16: Jogger s Oxygen Uptake A physician claims that joggers maximal volume oxygen uptake is greater than the average of all adults. A sample of 15 joggers has a mean of 40.6 milliliters per kilogram (ml/kg) and a standard deviation of 6 ml/kg. If the average of all adults is 36.7 ml/kg, is there enough evidence to support the physician s claim at α = 0.05? Step 1: State the hypotheses and identify the claim. H 0 : μ = 36.7 and H 1 : μ > 36.7 (claim) Step 2: Compute the test value. t X s n 6 15 Example 8-16: Jogger s Oxygen Uptake A physician claims that joggers maximal volume oxygen uptake is greater than the average of all adults. A sample of 15 joggers has a mean of 40.6 milliliters per kilogram (ml/kg) and a standard deviation of 6 ml/kg. If the average of all adults is 36.7 ml/kg, is there enough evidence to support the physician s claim at α = 0.05? Step 3: Find the P-value. In the d.f. = 14 row, falls between and 2.624, corresponding to α = and α = Thus, the P-value is somewhere between 0.01 and 0.025, since this is a one-tailed test. Bluman, 75 Bluman, 76 Example 8-16: Jogger s Oxygen Uptake The decision is to reject the null hypothesis, since the P-value < Whether to use z or t There is enough evidence to support the claim that the joggers maximal volume oxygen uptake is greater than 36.7 ml/kg. Bluman, 77 Bluman, 78 13

14 8.4 z Test for a Proportion Since a normal distribution can be used to approximate the binomial distribution when np 5 and nq 5, the standard normal distribution can be used to test hypotheses for proportions. The formula for the z test for a proportion is ˆ z p p pq n where X pˆ sample proportion n p population proportion n sample size Section 8-4 Example 8-17 Page #436 Bluman, 79 Bluman, 80 Example 8-17: Avoiding Trans Fats A dietician claims that 60% of people are trying to avoid trans fats in their diets. She randomly selected 200 people and found that 128 people stated that they were trying to avoid trans fats in their diets. At α = 0.05, is there enough evidence to reject the dietitian s claim? Step 1: State t the hypotheses and identify the claim. H 0 : p = 0.60 (claim) and H 1 : p 0.60 Step 2: Find the critical value. Since α = 0.05 and the test is a two-tailed test, the critical value is z = ±1.96. Example 8-17: Avoiding Trans Fats A dietician claims that 60% of people are trying to avoid trans fats in their diets. She randomly selected 200 people and found that 128 people stated that they were trying to avoid trans fats in their diets. At α = 0.05, is there enough evidence to reject the dietitian s claim? Step 3: Compute the test t value. X 128 pˆ 0.64 n 200 ˆ z p p pq n Bluman, 81 Bluman, 82 Example 8-17: Avoiding Trans Fats Do not reject the null hypothesis since the test value falls outside the critical region. There is not enough evidence to reject the claim that 60% of people are trying to avoid trans fats in their diets. Section 8-4 Example 8-18 Page #437 Bluman, 83 Bluman, 84 14

15 Example 8-18: Family/Medical Leave Act The Family and Medical Leave Act provides job protection and unpaid time off from work for a serious illness or birth of a child. Example 8-18: Family/Medical Leave Act Step 1 State the hypotheses and identify the claim. In 2000, 60% of the respondents of a survey stated that it was very easy to get time off for these circumstances. A researcher wishes to see if the percentage who said that it was very easy to get time off has changed. A sample of 100 people who used the leave said that 53% found it easy to use the leave. At α = 0.01, has the percentage changed? Step 2 Step 3 Find the critical value(s). Since α = 0.01 and this test is two-tailed, the critical values are ±2.58. Compute the test value. It is not necessary to find ˆp since it is given in the exercise; pˆ 53%. Substitute in the formula and evaluate. Bluman, 85 Bluman, 86 Example 8-18: Family/Medical Leave Act Step 4 Make the decision. Do not reject the null hypothesis, since the test value falls in the noncritical region. Example 8-18: Family/Medical Leave Act Step 5 Summarize the results. There is not enough evidence to support the claim that the percentage of those using the medical leave said that it was easy to get has changed. Bluman, 87 Bluman, Confidence Intervals and There is a relationship between confidence intervals and hypothesis testing. When the null hypothesis is rejected in a hypothesis- testing situation, the confidence interval for the mean using the same level of significance will not contain the hypothesized mean. Likewise, when the null hypothesis is not rejected, the confidence interval computed using the same level of significance will contain the hypothesized mean. Section 8-6 Example 8-30 Page #455 Bluman, 89 Bluman, 90 15

16 Example 8-30: Sugar Production Sugar is packed in 5-pound bags. An inspector suspects the bags may not contain 5 pounds. A sample of 50 bags produces a mean of 4.6 pounds and a standard deviation of 0.7 pound. Is there enough evidence to conclude that the bags do not contain 5 pounds as stated at α = 0.05? Also, find the 95% confidence interval of the true mean. Step 1: State the hypotheses and identify the claim. H 0 : μ = 5 and H 1 : μ 5 (claim) Step 2: Find the critical values. The critical values are t = ± Example 8-30: Sugar Production Sugar is packed in 5-pound bags. An inspector suspects the bags may not contain 5 pounds. A sample of 50 bags produces a mean of 4.6 pounds and a standard deviation of 0.7 pound. Is there enough evidence to conclude that the bags do not contain 5 pounds as stated at α = 0.05? Also, find the 95% confidence interval of the true mean. Step 3: Compute the test value. X t 4.04 s n Bluman, 91 Bluman, 92 Example 8-30: Sugar Production Since , the null hypothesis is rejected. Example 8-30: Sugar Production The 95% confidence interval for the mean is There is enough evidence to support the claim that the bags do not weigh 5 pounds. Notice that the 95% confidence interval of m does not contain the hypothesized value μ = 5. Hence, there is agreement between the hypothesis test and the confidence interval. Bluman, 93 Bluman, 94 16

### 8 6 X 2 Test for a Variance or Standard Deviation

Section 8 6 x 2 Test for a Variance or Standard Deviation 437 This test uses the P-value method. Therefore, it is not necessary to enter a significance level. 1. Select MegaStat>Hypothesis Tests>Proportion

### Chapter 8: Hypothesis Testing for One Population Mean, Variance, and Proportion

Chapter 8: Hypothesis Testing for One Population Mean, Variance, and Proportion Learning Objectives Upon successful completion of Chapter 8, you will be able to: Understand terms. State the null and alternative

### HYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1. used confidence intervals to answer questions such as...

HYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1 PREVIOUSLY used confidence intervals to answer questions such as... You know that 0.25% of women have red/green color blindness. You conduct a study of men

### Hypothesis Testing --- One Mean

Hypothesis Testing --- One Mean A hypothesis is simply a statement that something is true. Typically, there are two hypotheses in a hypothesis test: the null, and the alternative. Null Hypothesis The hypothesis

### 1. What is the critical value for this 95% confidence interval? CV = z.025 = invnorm(0.025) = 1.96

1 Final Review 2 Review 2.1 CI 1-propZint Scenario 1 A TV manufacturer claims in its warranty brochure that in the past not more than 10 percent of its TV sets needed any repair during the first two years

### Chapter 8 Hypothesis Testing Chapter 8 Hypothesis Testing 8-1 Overview 8-2 Basics of Hypothesis Testing

Chapter 8 Hypothesis Testing 1 Chapter 8 Hypothesis Testing 8-1 Overview 8-2 Basics of Hypothesis Testing 8-3 Testing a Claim About a Proportion 8-5 Testing a Claim About a Mean: s Not Known 8-6 Testing

### Two Related Samples t Test

Two Related Samples t Test In this example 1 students saw five pictures of attractive people and five pictures of unattractive people. For each picture, the students rated the friendliness of the person

### Section 7.1. Introduction to Hypothesis Testing. Schrodinger s cat quantum mechanics thought experiment (1935)

Section 7.1 Introduction to Hypothesis Testing Schrodinger s cat quantum mechanics thought experiment (1935) Statistical Hypotheses A statistical hypothesis is a claim about a population. Null hypothesis

### Stats Review Chapters 9-10

Stats Review Chapters 9-10 Created by Teri Johnson Math Coordinator, Mary Stangler Center for Academic Success Examples are taken from Statistics 4 E by Michael Sullivan, III And the corresponding Test

### 3.4 Statistical inference for 2 populations based on two samples

3.4 Statistical inference for 2 populations based on two samples Tests for a difference between two population means The first sample will be denoted as X 1, X 2,..., X m. The second sample will be denoted

### An Introduction to Statistics Course (ECOE 1302) Spring Semester 2011 Chapter 10- TWO-SAMPLE TESTS

The Islamic University of Gaza Faculty of Commerce Department of Economics and Political Sciences An Introduction to Statistics Course (ECOE 130) Spring Semester 011 Chapter 10- TWO-SAMPLE TESTS Practice

### HYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1. used confidence intervals to answer questions such as...

HYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1 PREVIOUSLY used confidence intervals to answer questions such as... You know that 0.25% of women have red/green color blindness. You conduct a study of men

### HYPOTHESIS TESTING: POWER OF THE TEST

HYPOTHESIS TESTING: POWER OF THE TEST The first 6 steps of the 9-step test of hypothesis are called "the test". These steps are not dependent on the observed data values. When planning a research project,

### 9 Testing the Difference

blu49076_ch09.qxd 5/1/2003 8:19 AM Page 431 c h a p t e r 9 9 Testing the Difference Between Two Means, Two Variances, and Two Proportions Outline 9 1 Introduction 9 2 Testing the Difference Between Two

### Introduction to Hypothesis Testing

I. Terms, Concepts. Introduction to Hypothesis Testing A. In general, we do not know the true value of population parameters - they must be estimated. However, we do have hypotheses about what the true

### Math 251, Review Questions for Test 3 Rough Answers

Math 251, Review Questions for Test 3 Rough Answers 1. (Review of some terminology from Section 7.1) In a state with 459,341 voters, a poll of 2300 voters finds that 45 percent support the Republican candidate,

### Hypothesis testing - Steps

Hypothesis testing - Steps Steps to do a two-tailed test of the hypothesis that β 1 0: 1. Set up the hypotheses: H 0 : β 1 = 0 H a : β 1 0. 2. Compute the test statistic: t = b 1 0 Std. error of b 1 =

### Final Exam Practice Problem Answers

Final Exam Practice Problem Answers The following data set consists of data gathered from 77 popular breakfast cereals. The variables in the data set are as follows: Brand: The brand name of the cereal

### Regression Analysis: A Complete Example

Regression Analysis: A Complete Example This section works out an example that includes all the topics we have discussed so far in this chapter. A complete example of regression analysis. PhotoDisc, Inc./Getty

### Section 8-1 Pg. 410 Exercises 12,13

Section 8- Pg. 4 Exercises 2,3 2. Using the z table, find the critical value for each. a) α=.5, two-tailed test, answer: -.96,.96 b) α=., left-tailed test, answer: -2.33, 2.33 c) α=.5, right-tailed test,

### LAB 4 INSTRUCTIONS CONFIDENCE INTERVALS AND HYPOTHESIS TESTING

LAB 4 INSTRUCTIONS CONFIDENCE INTERVALS AND HYPOTHESIS TESTING In this lab you will explore the concept of a confidence interval and hypothesis testing through a simulation problem in engineering setting.

### A POPULATION MEAN, CONFIDENCE INTERVALS AND HYPOTHESIS TESTING

CHAPTER 5. A POPULATION MEAN, CONFIDENCE INTERVALS AND HYPOTHESIS TESTING 5.1 Concepts When a number of animals or plots are exposed to a certain treatment, we usually estimate the effect of the treatment

### Online 12 - Sections 9.1 and 9.2-Doug Ensley

Student: Date: Instructor: Doug Ensley Course: MAT117 01 Applied Statistics - Ensley Assignment: Online 12 - Sections 9.1 and 9.2 1. Does a P-value of 0.001 give strong evidence or not especially strong

### C. The null hypothesis is not rejected when the alternative hypothesis is true. A. population parameters.

Sample Multiple Choice Questions for the material since Midterm 2. Sample questions from Midterms and 2 are also representative of questions that may appear on the final exam.. A randomly selected sample

### BA 275 Review Problems - Week 5 (10/23/06-10/27/06) CD Lessons: 48, 49, 50, 51, 52 Textbook: pp. 380-394

BA 275 Review Problems - Week 5 (10/23/06-10/27/06) CD Lessons: 48, 49, 50, 51, 52 Textbook: pp. 380-394 1. Does vigorous exercise affect concentration? In general, the time needed for people to complete

### Chapter Study Guide. Chapter 11 Confidence Intervals and Hypothesis Testing for Means

OPRE504 Chapter Study Guide Chapter 11 Confidence Intervals and Hypothesis Testing for Means I. Calculate Probability for A Sample Mean When Population σ Is Known 1. First of all, we need to find out the

### BA 275 Review Problems - Week 6 (10/30/06-11/3/06) CD Lessons: 53, 54, 55, 56 Textbook: pp. 394-398, 404-408, 410-420

BA 275 Review Problems - Week 6 (10/30/06-11/3/06) CD Lessons: 53, 54, 55, 56 Textbook: pp. 394-398, 404-408, 410-420 1. Which of the following will increase the value of the power in a statistical test

### MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

STT315 Practice Ch 5-7 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 1) The length of time a traffic signal stays green (nicknamed

### Hypothesis Testing: Two Means, Paired Data, Two Proportions

Chapter 10 Hypothesis Testing: Two Means, Paired Data, Two Proportions 10.1 Hypothesis Testing: Two Population Means and Two Population Proportions 1 10.1.1 Student Learning Objectives By the end of this

### In the past, the increase in the price of gasoline could be attributed to major national or global

Chapter 7 Testing Hypotheses Chapter Learning Objectives Understanding the assumptions of statistical hypothesis testing Defining and applying the components in hypothesis testing: the research and null

### Name: (b) Find the minimum sample size you should use in order for your estimate to be within 0.03 of p when the confidence level is 95%.

Chapter 7-8 Exam Name: Answer the questions in the spaces provided. If you run out of room, show your work on a separate paper clearly numbered and attached to this exam. Please indicate which program

### THE FIRST SET OF EXAMPLES USE SUMMARY DATA... EXAMPLE 7.2, PAGE 227 DESCRIBES A PROBLEM AND A HYPOTHESIS TEST IS PERFORMED IN EXAMPLE 7.

THERE ARE TWO WAYS TO DO HYPOTHESIS TESTING WITH STATCRUNCH: WITH SUMMARY DATA (AS IN EXAMPLE 7.17, PAGE 236, IN ROSNER); WITH THE ORIGINAL DATA (AS IN EXAMPLE 8.5, PAGE 301 IN ROSNER THAT USES DATA FROM

### 7 Confidence Intervals

blu49076_ch07.qxd 5/20/2003 3:15 PM Page 325 c h a p t e r 7 7 Confidence Intervals and Sample Size Outline 7 1 Introduction 7 2 Confidence Intervals for the Mean (s Known or n 30) and Sample Size 7 3

### Chapter 7: One-Sample Inference

Chapter 7: One-Sample Inference Now that you have all this information about descriptive statistics and probabilities, it is time to start inferential statistics. There are two branches of inferential

### WISE Power Tutorial All Exercises

ame Date Class WISE Power Tutorial All Exercises Power: The B.E.A.. Mnemonic Four interrelated features of power can be summarized using BEA B Beta Error (Power = 1 Beta Error): Beta error (or Type II

### SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Ch. 10 Chi SquareTests and the F-Distribution 10.1 Goodness of Fit 1 Find Expected Frequencies Provide an appropriate response. 1) The frequency distribution shows the ages for a sample of 100 employees.

### Introduction to. Hypothesis Testing CHAPTER LEARNING OBJECTIVES. 1 Identify the four steps of hypothesis testing.

Introduction to Hypothesis Testing CHAPTER 8 LEARNING OBJECTIVES After reading this chapter, you should be able to: 1 Identify the four steps of hypothesis testing. 2 Define null hypothesis, alternative

### t Tests in Excel The Excel Statistical Master By Mark Harmon Copyright 2011 Mark Harmon

t-tests in Excel By Mark Harmon Copyright 2011 Mark Harmon No part of this publication may be reproduced or distributed without the express permission of the author. mark@excelmasterseries.com www.excelmasterseries.com

### November 08, 2010. 155S8.6_3 Testing a Claim About a Standard Deviation or Variance

Chapter 8 Hypothesis Testing 8 1 Review and Preview 8 2 Basics of Hypothesis Testing 8 3 Testing a Claim about a Proportion 8 4 Testing a Claim About a Mean: σ Known 8 5 Testing a Claim About a Mean: σ

### Hypothesis Testing. Hypothesis Testing

Hypothesis Testing Daniel A. Menascé Department of Computer Science George Mason University 1 Hypothesis Testing Purpose: make inferences about a population parameter by analyzing differences between observed

Chapter 11 Testing Hypotheses About Proportions Hypothesis testing method: uses data from a sample to judge whether or not a statement about a population may be true. Steps in Any Hypothesis Test 1. Determine

### Review #2. Statistics

Review #2 Statistics Find the mean of the given probability distribution. 1) x P(x) 0 0.19 1 0.37 2 0.16 3 0.26 4 0.02 A) 1.64 B) 1.45 C) 1.55 D) 1.74 2) The number of golf balls ordered by customers of

### FAT-FREE OR REGULAR PRINGLES: CAN TASTERS TELL THE DIFFERENCE?

CHAPTER 10 Hypothesis Tests Involving a Sample Mean or Proportion FAT-FREE OR REGULAR PRINGLES: CAN TASTERS TELL THE DIFFERENCE? Michael Newman/PhotoEdit When the makers of Pringles potato chips came out

### Chapter 2. Hypothesis testing in one population

Chapter 2. Hypothesis testing in one population Contents Introduction, the null and alternative hypotheses Hypothesis testing process Type I and Type II errors, power Test statistic, level of significance

### Unit 26 Estimation with Confidence Intervals

Unit 26 Estimation with Confidence Intervals Objectives: To see how confidence intervals are used to estimate a population proportion, a population mean, a difference in population proportions, or a difference

### Confidence intervals

Confidence intervals Today, we re going to start talking about confidence intervals. We use confidence intervals as a tool in inferential statistics. What this means is that given some sample statistics,

### STAT 350 Practice Final Exam Solution (Spring 2015)

PART 1: Multiple Choice Questions: 1) A study was conducted to compare five different training programs for improving endurance. Forty subjects were randomly divided into five groups of eight subjects

### Practice problems for Homework 12 - confidence intervals and hypothesis testing. Open the Homework Assignment 12 and solve the problems.

Practice problems for Homework 1 - confidence intervals and hypothesis testing. Read sections 10..3 and 10.3 of the text. Solve the practice problems below. Open the Homework Assignment 1 and solve the

### Experimental Design. Power and Sample Size Determination. Proportions. Proportions. Confidence Interval for p. The Binomial Test

Experimental Design Power and Sample Size Determination Bret Hanlon and Bret Larget Department of Statistics University of Wisconsin Madison November 3 8, 2011 To this point in the semester, we have largely

### Introduction to Hypothesis Testing OPRE 6301

Introduction to Hypothesis Testing OPRE 6301 Motivation... The purpose of hypothesis testing is to determine whether there is enough statistical evidence in favor of a certain belief, or hypothesis, about

### Difference of Means and ANOVA Problems

Difference of Means and Problems Dr. Tom Ilvento FREC 408 Accounting Firm Study An accounting firm specializes in auditing the financial records of large firm It is interested in evaluating its fee structure,particularly

### General Method: Difference of Means. 3. Calculate df: either Welch-Satterthwaite formula or simpler df = min(n 1, n 2 ) 1.

General Method: Difference of Means 1. Calculate x 1, x 2, SE 1, SE 2. 2. Combined SE = SE1 2 + SE2 2. ASSUMES INDEPENDENT SAMPLES. 3. Calculate df: either Welch-Satterthwaite formula or simpler df = min(n

### Calculating P-Values. Parkland College. Isela Guerra Parkland College. Recommended Citation

Parkland College A with Honors Projects Honors Program 2014 Calculating P-Values Isela Guerra Parkland College Recommended Citation Guerra, Isela, "Calculating P-Values" (2014). A with Honors Projects.

### 6: Introduction to Hypothesis Testing

6: Introduction to Hypothesis Testing Significance testing is used to help make a judgment about a claim by addressing the question, Can the observed difference be attributed to chance? We break up significance

### Comparing Two Groups. Standard Error of ȳ 1 ȳ 2. Setting. Two Independent Samples

Comparing Two Groups Chapter 7 describes two ways to compare two populations on the basis of independent samples: a confidence interval for the difference in population means and a hypothesis test. The

### Principles of Hypothesis Testing for Public Health

Principles of Hypothesis Testing for Public Health Laura Lee Johnson, Ph.D. Statistician National Center for Complementary and Alternative Medicine johnslau@mail.nih.gov Fall 2011 Answers to Questions

### Module 2 Probability and Statistics

Module 2 Probability and Statistics BASIC CONCEPTS Multiple Choice Identify the choice that best completes the statement or answers the question. 1. The standard deviation of a standard normal distribution

### Mind on Statistics. Chapter 13

Mind on Statistics Chapter 13 Sections 13.1-13.2 1. Which statement is not true about hypothesis tests? A. Hypothesis tests are only valid when the sample is representative of the population for the question

### Name: Date: Use the following to answer questions 3-4:

Name: Date: 1. Determine whether each of the following statements is true or false. A) The margin of error for a 95% confidence interval for the mean increases as the sample size increases. B) The margin

### Recall this chart that showed how most of our course would be organized:

Chapter 4 One-Way ANOVA Recall this chart that showed how most of our course would be organized: Explanatory Variable(s) Response Variable Methods Categorical Categorical Contingency Tables Categorical

### Chapter 7 TEST OF HYPOTHESIS

Chapter 7 TEST OF HYPOTHESIS In a certain perspective, we can view hypothesis testing just like a jury in a court trial. In a jury trial, the null hypothesis is similar to the jury making a decision of

### STAT 145 (Notes) Al Nosedal anosedal@unm.edu Department of Mathematics and Statistics University of New Mexico. Fall 2013

STAT 145 (Notes) Al Nosedal anosedal@unm.edu Department of Mathematics and Statistics University of New Mexico Fall 2013 CHAPTER 18 INFERENCE ABOUT A POPULATION MEAN. Conditions for Inference about mean

### 5/31/2013. 6.1 Normal Distributions. Normal Distributions. Chapter 6. Distribution. The Normal Distribution. Outline. Objectives.

The Normal Distribution C H 6A P T E R The Normal Distribution Outline 6 1 6 2 Applications of the Normal Distribution 6 3 The Central Limit Theorem 6 4 The Normal Approximation to the Binomial Distribution

### Lesson 1: Comparison of Population Means Part c: Comparison of Two- Means

Lesson : Comparison of Population Means Part c: Comparison of Two- Means Welcome to lesson c. This third lesson of lesson will discuss hypothesis testing for two independent means. Steps in Hypothesis

### Introduction to Hypothesis Testing. Hypothesis Testing. Step 1: State the Hypotheses

Introduction to Hypothesis Testing 1 Hypothesis Testing A hypothesis test is a statistical procedure that uses sample data to evaluate a hypothesis about a population Hypothesis is stated in terms of the

### Simple Regression Theory II 2010 Samuel L. Baker

SIMPLE REGRESSION THEORY II 1 Simple Regression Theory II 2010 Samuel L. Baker Assessing how good the regression equation is likely to be Assignment 1A gets into drawing inferences about how close the

### Using Excel for inferential statistics

FACT SHEET Using Excel for inferential statistics Introduction When you collect data, you expect a certain amount of variation, just caused by chance. A wide variety of statistical tests can be applied

### Summary of Formulas and Concepts. Descriptive Statistics (Ch. 1-4)

Summary of Formulas and Concepts Descriptive Statistics (Ch. 1-4) Definitions Population: The complete set of numerical information on a particular quantity in which an investigator is interested. We assume

### 4. Continuous Random Variables, the Pareto and Normal Distributions

4. Continuous Random Variables, the Pareto and Normal Distributions A continuous random variable X can take any value in a given range (e.g. height, weight, age). The distribution of a continuous random

### Part 3. Comparing Groups. Chapter 7 Comparing Paired Groups 189. Chapter 8 Comparing Two Independent Groups 217

Part 3 Comparing Groups Chapter 7 Comparing Paired Groups 189 Chapter 8 Comparing Two Independent Groups 217 Chapter 9 Comparing More Than Two Groups 257 188 Elementary Statistics Using SAS Chapter 7 Comparing

### Estimation of σ 2, the variance of ɛ

Estimation of σ 2, the variance of ɛ The variance of the errors σ 2 indicates how much observations deviate from the fitted surface. If σ 2 is small, parameters β 0, β 1,..., β k will be reliably estimated

### 5.1 Identifying the Target Parameter

University of California, Davis Department of Statistics Summer Session II Statistics 13 August 20, 2012 Date of latest update: August 20 Lecture 5: Estimation with Confidence intervals 5.1 Identifying

### Testing a claim about a population mean

Introductory Statistics Lectures Testing a claim about a population mean One sample hypothesis test of the mean Department of Mathematics Pima Community College Redistribution of this material is prohibited

### DDBA 8438: Introduction to Hypothesis Testing Video Podcast Transcript

DDBA 8438: Introduction to Hypothesis Testing Video Podcast Transcript JENNIFER ANN MORROW: Welcome to "Introduction to Hypothesis Testing." My name is Dr. Jennifer Ann Morrow. In today's demonstration,

### Comparing Multiple Proportions, Test of Independence and Goodness of Fit

Comparing Multiple Proportions, Test of Independence and Goodness of Fit Content Testing the Equality of Population Proportions for Three or More Populations Test of Independence Goodness of Fit Test 2

### NCSS Statistical Software. One-Sample T-Test

Chapter 205 Introduction This procedure provides several reports for making inference about a population mean based on a single sample. These reports include confidence intervals of the mean or median,

### Business Statistics, 9e (Groebner/Shannon/Fry) Chapter 9 Introduction to Hypothesis Testing

Business Statistics, 9e (Groebner/Shannon/Fry) Chapter 9 Introduction to Hypothesis Testing 1) Hypothesis testing and confidence interval estimation are essentially two totally different statistical procedures

### 1) The table lists the smoking habits of a group of college students. Answer: 0.218

FINAL EXAM REVIEW Name ) The table lists the smoking habits of a group of college students. Sex Non-smoker Regular Smoker Heavy Smoker Total Man 5 52 5 92 Woman 8 2 2 220 Total 22 2 If a student is chosen

### Chapter 4. iclicker Question 4.4 Pre-lecture. Part 2. Binomial Distribution. J.C. Wang. iclicker Question 4.4 Pre-lecture

Chapter 4 Part 2. Binomial Distribution J.C. Wang iclicker Question 4.4 Pre-lecture iclicker Question 4.4 Pre-lecture Outline Computing Binomial Probabilities Properties of a Binomial Distribution Computing

### Practice Problems and Exams

Practice Problems and Exams 1 The Islamic University of Gaza Faculty of Commerce Department of Economics and Political Sciences An Introduction to Statistics Course (ECOE 1302) Spring Semester 2009-2010

### 22. HYPOTHESIS TESTING

22. HYPOTHESIS TESTING Often, we need to make decisions based on incomplete information. Do the data support some belief ( hypothesis ) about the value of a population parameter? Is OJ Simpson guilty?

### Mind on Statistics. Chapter 12

Mind on Statistics Chapter 12 Sections 12.1 Questions 1 to 6: For each statement, determine if the statement is a typical null hypothesis (H 0 ) or alternative hypothesis (H a ). 1. There is no difference

### Tests for One Proportion

Chapter 100 Tests for One Proportion Introduction The One-Sample Proportion Test is used to assess whether a population proportion (P1) is significantly different from a hypothesized value (P0). This is

### Descriptive and Inferential Statistics

General Sir John Kotelawala Defence University Workshop on Descriptive and Inferential Statistics Faculty of Research and Development 14 th May 2013 1. Introduction to Statistics 1.1 What is Statistics?

### Unit 31 A Hypothesis Test about Correlation and Slope in a Simple Linear Regression

Unit 31 A Hypothesis Test about Correlation and Slope in a Simple Linear Regression Objectives: To perform a hypothesis test concerning the slope of a least squares line To recognize that testing for a

### Two-Sample T-Tests Assuming Equal Variance (Enter Means)

Chapter 4 Two-Sample T-Tests Assuming Equal Variance (Enter Means) Introduction This procedure provides sample size and power calculations for one- or two-sided two-sample t-tests when the variances of

### Descriptive Statistics

Descriptive Statistics Primer Descriptive statistics Central tendency Variation Relative position Relationships Calculating descriptive statistics Descriptive Statistics Purpose to describe or summarize

### Math 108 Exam 3 Solutions Spring 00

Math 108 Exam 3 Solutions Spring 00 1. An ecologist studying acid rain takes measurements of the ph in 12 randomly selected Adirondack lakes. The results are as follows: 3.0 6.5 5.0 4.2 5.5 4.7 3.4 6.8

### Lecture Notes Module 1

Lecture Notes Module 1 Study Populations A study population is a clearly defined collection of people, animals, plants, or objects. In psychological research, a study population usually consists of a specific

### Ch. 6.1 #7-49 odd. The area is found by looking up z= 0.75 in Table E and subtracting 0.5. Area = 0.7734-0.5= 0.2734

Ch. 6.1 #7-49 odd The area is found by looking up z= 0.75 in Table E and subtracting 0.5. Area = 0.7734-0.5= 0.2734 The area is found by looking up z= 2.07 in Table E and subtracting from 0.5. Area = 0.5-0.0192

### " Y. Notation and Equations for Regression Lecture 11/4. Notation:

Notation: Notation and Equations for Regression Lecture 11/4 m: The number of predictor variables in a regression Xi: One of multiple predictor variables. The subscript i represents any number from 1 through

### 2 Precision-based sample size calculations

Statistics: An introduction to sample size calculations Rosie Cornish. 2006. 1 Introduction One crucial aspect of study design is deciding how big your sample should be. If you increase your sample size

### How To Test For Significance On A Data Set

Non-Parametric Univariate Tests: 1 Sample Sign Test 1 1 SAMPLE SIGN TEST A non-parametric equivalent of the 1 SAMPLE T-TEST. ASSUMPTIONS: Data is non-normally distributed, even after log transforming.

### Simple Linear Regression Inference

Simple Linear Regression Inference 1 Inference requirements The Normality assumption of the stochastic term e is needed for inference even if it is not a OLS requirement. Therefore we have: Interpretation

### Review. March 21, 2011. 155S7.1 2_3 Estimating a Population Proportion. Chapter 7 Estimates and Sample Sizes. Test 2 (Chapters 4, 5, & 6) Results

MAT 155 Statistical Analysis Dr. Claude Moore Cape Fear Community College Chapter 7 Estimates and Sample Sizes 7 1 Review and Preview 7 2 Estimating a Population Proportion 7 3 Estimating a Population

### Two-sample hypothesis testing, II 9.07 3/16/2004

Two-sample hypothesis testing, II 9.07 3/16/004 Small sample tests for the difference between two independent means For two-sample tests of the difference in mean, things get a little confusing, here,

### Permutation Tests for Comparing Two Populations

Permutation Tests for Comparing Two Populations Ferry Butar Butar, Ph.D. Jae-Wan Park Abstract Permutation tests for comparing two populations could be widely used in practice because of flexibility of