Statistical Inference. Section 9.1 Significance Tests: The Basics. Significance Test. The Reasoning of Significance Tests.

Transcription

1 Section 9.1 Significance Tests: The Basics Significance Test A significance test is a formal procedure for comparing observed data with a claim (also called a hypothesis) whose truth we want to assess. The claim is a statement about a parameter, like the population proportion p or the population mean µ. We express the results of a significance test in terms of a probability (p value) that measures how well the data and the claim agree. Section 9.1 Significance Tests: The Basics After this section, you should be able to STATE correct hypotheses for a significance test about a population proportion or mean. INTERPRET P values in context. INTERPRET a Type I error and a Type II error in context, and give the consequences of each. DESCRIBE the relationship between the significance level of a test, P(Type II error), and power. The Reasoning of Significance Tests Statistical tests deal with claims about a population. Tests ask if sample data give good evidence against a claim. A test might say, If we took many random samples and the claim were true, what is the probability we will get a result like this. For example: Suppose a basketball player claimed to be an 80% free throw shooter. To test this claim, we have him attempt 50 free throws. He makes 32 of them. His sample proportion of made shots is 32/50 = What can we conclude about the claim based on this sample data? What is the probability the player is telling the truth?!?! Statistical Inference The Reasoning of Significance Tests We can use software to simulate 400 sets of 50 shots assuming that the player is really an 80% shooter. Significance Tests ASSESS the evidence provided by data about some claim concerning a population Reject or fail to reject (Yes vs. No) Confidence Interval ESTIMATE a population parameter. Give range of possible values 1

2 The Reasoning of Significance Tests You can say how strong the evidence against the player s claim is by giving the probability that he would make as few as 32 out of 50 free throws if he really makes 80% in the long run. The observed statistic is so unlikely if the actual parameter value is p = 0.80 that it gives convincing evidence that the player s claim is not true. Stating Hypotheses In any significance test, the null hypothesis has the form H 0 : parameter = value The alternative hypothesis has one of the forms H a : parameter < value H a : parameter > value H a : parameter value To determine the correct form of H a, read the problem carefully. The Reasoning of Significance Tests Based on the evidence, we might conclude the player s claim is incorrect. In reality, there are two possible explanations for the fact that he made only 64% of his free throws. 1) The player s claim is correct (p = 0.8), and by horrible luck, a very unlikely outcome occurred. 2) The population proportion is actually less than 0.8, so the sample result is not an unlikely outcome. Basic Idea An outcome that would rarely happen if a claim were true is good evidence that the claim is not true. Stating Hypotheses The alternative hypothesis is one sided if it states that a parameter is larger than the null hypothesis value or if it states that the parameter is smaller than the null value. It is two sided if it states that the parameter is different from the null hypothesis value (it could be either larger or smaller). Use H a : parameter value for two sided. Hypotheses always refer to a population, not to a sample. Be sure to state H 0 and H a in terms of population parameters (p or ). It is never correct to write a hypothesis about a sample statistic, such as Stating Hypotheses The claim tested by a statistical test is called the null hypothesis (H 0 ). The test is designed to assess the strength of the evidence against the null hypothesis. Often the null hypothesis is a statement of no difference or that the claim is true. The claim about the population that we are trying to find evidence for is the alternative hypothesis (H a ). State the Hypothesis & Parameter: A high school junior running for student body president, Sally, claims that 80% of the student body favors her in the school election. Her opponent believes this percentage to be lower, write the appropriate null and alternative hypotheses. In the free throw shooter example, our hypotheses are H 0 : p = 0.80 H a : p < 0.80 Parameter: p = the long run proportion of made free throws. 2

3 State the Hypothesis & Parameter: p= true proportion of students that favor Sally for president. Example: Studying Job Satisfaction Does the job satisfaction of assembly line workers differ when their work is machine paced rather than self paced? One study chose 18 subjects at random from a company with over 200 workers who assembled electronic devices. Half of the workers were assigned at random to each of two groups. Both groups did similar assembly work, but one group was allowed to pace themselves while the other group used an assembly line that moved at a fixed pace. After two weeks, all the workers took a test of job satisfaction. Then they switched work setups and took the test again after two more weeks. The response variable is the difference in satisfaction scores, self paced minus machine paced. a) Describe the parameter of interest in this setting. b) State appropriate hypotheses for performing a significance test. Interpreting P Values The null hypothesis H 0 states the claim that we are seeking evidence against. The probability that measures the strength of the evidence against a null hypothesis is called a P value. The probability, computed assuming H 0 is true, that the statistic would take a value as extreme as or more extreme than the one actually observed is called the P value of the test. The smaller the P value, the stronger the evidence against H 0 provided by the data. Example: Studying Job Satisfaction a) Describe the parameter of interest in this setting. The parameter of interest is the mean µ of the differences (self paced minus machine paced) in job satisfaction scores in the population of all assembly line workers at this company. (FYI Matched pairs!!!) b) State appropriate hypotheses for performing a significance test. Because the initial question asked whether job satisfaction differs, the alternative hypothesis is two sided; that is, either µ < 0 or µ > 0. For simplicity, we write this as µ 0. That is, H 0 : µ = 0 H a : µ 0 H 0 : µ = 0 English Math Example: Studying Job Satisfaction Small P value Evidence against Unlikely to occur if H 0 is true We reject H 0 For the job satisfaction study, the hypotheses are H 0 : µ = 0 H a : µ 0 Large P value Not convincing evidence Could occur if H 0 is true We fail to reject H 0 c) Explain what it means for the null hypothesis to be true in this setting. d) Interpret the given P value in context. 3

4 Example: Studying Job Satisfaction For the job satisfaction study, the hypotheses are H 0 : µ = 0 H a : µ 0 c) Explain what it means for the null hypothesis to be true in this setting. In this setting, the null hypothesis is that there is no mean difference in employee satisfaction scores (self paced machine paced) for the entire population of assemblyline workers at the company. If null hypothesis is true, then the workers don t favor one work environment over the other, on average. Statistically Significant There is no perfect rule for how small a P value we should require in order to reject H 0 it s a matter of judgment and depends on the specific circumstances. We can compare the P value with a fixed value (typically α = 0.05), called the significance level (alpha α). When our p value is greater than the chosen α, there is no statistically significance. We fail to reject the null. When our P value is less than the chosen α, we say that the result is statistically significant. In that case, we reject the null hypothesis H 0 and conclude that there is convincing evidence in favor of the alternative hypothesis H a. Example: Studying Job Satisfaction d) Interpret the P-value in context. The P value is the probability of observing a sample result as extreme as (or more extreme) by pure chance given that the null hypothesis is actually true. General conclusion in a significance test : P value small reject H 0 conclude H a P value large fail to reject H 0 cannot conclude H a Since the test is two sided, we have a 23% chance of observing a value that is 17 points or more from the mean, in either direction. An outcome that would occur so often just by chance (almost 1 in every 4 random samples of 18 workers) when the null is true is not convincing evidence against null. We fail to reject H 0 : µ = 0. Conclusion with fixed level of significance : P value < α reject H 0 conclude H a P value α fail to reject H 0 cannot conclude H a Conclusion: Statistical Significance We will make one of two decisions based on the strength of the evidence against the null hypothesis (and in favor of the alternative hypothesis) reject H 0 or fail to reject H 0. If our sample result is too unlikely to have happened by chance assuming H 0 is true, then we ll reject H 0. Otherwise, we will fail to reject H 0. A fail to reject H 0 decision in a significance test doesn t mean that H 0 is true. For that reason, you should never accept H 0 or use language implying that you believe H 0 is true. Example: Better Batteries A company has developed a new deluxe AAA battery that is supposed to last longer than its regular AAA battery. However, these new batteries are more expensive to produce, so the company would like to be convinced that they really do last longer. Based on years of experience, the company knows that its regular AAA batteries last for 30 hours of continuous use, on average. The company selects an SRS of 15 new batteries and uses them continuously until they are completely drained. A significance test is performed using the hypotheses H 0 : µ = 30 hours H a : µ > 30 hours where µ is the true mean lifetime of the new deluxe AAA batteries. The resulting P value is a) What conclusion can you make for the significance level α = 0.05? b) What conclusion can you make for the significance level α = 0.01? 4

5 Example: Better Batteries a) What conclusion can you make for the significance level α = 0.05? Since the P value, , is less than α = 0.05, the sample result is statistically significant at the 5% level. We have sufficient evidence to reject the null hypothesis and have sufficient evidence to conclude that the company s deluxe AAA batteries last longer than 30 hours, on average. b) What conclusion can you make at significance level α = 0.01? Since the P value, , is greater than α = 0.01, the sample result is not statistically significant at the 1% level. We fail to reject the null hypothesis; therefore, we cannot conclude that the deluxe AAA batteries last longer than 30 hours, on average. Type I & II Errors American Justice System Example: Ho: innocent Ha: guilty Type I error: punish an innocent person Type II error: let a not innocent (guilty) person go free More: Type I and Type II Errors Type I & II Errors Type I error Reject H 0 when H 0 is true Type II error Fail to reject H 0 when H 0 is false Double F = Type II Type I & II Errors Type I and II Errors Quality Control Example: Ho: the product is acceptable to the customer Ha: the product is unacceptable to the customer type I error: reject acceptable product and don't ship it. type 2 error: ship unacceptable product to the customer 5

6 Example: Perfect Potatoes A potato chip producer and its main supplier agree that each shipment of potatoes must meet certain quality standards. If the producer determines that more than 8% of the potatoes in the shipment have blemishes, the truck will be sent away to get another load of potatoes from the supplier. Otherwise, the entire truckload will be used to make potato chips. To make the decision, a supervisor will inspect a random sample of potatoes from the shipment. The producer will then perform a significance test using the hypotheses H 0 : p = 0.08 H a : p > 0.08 where p is the actual proportion of potatoes with blemishes in a given truckload. Power The probability of NOT making a Type II error. The higher the power, the less likely the mistake is. Describe a Type I and a Type II error in this setting, and explain the consequences of each. Example: Perfect Potatoes Describe a Type I and a Type II error in this setting, and explain the consequences of each. A Type I error would occur if the producer concludes that the proportion of potatoes with blemishes is greater than 0.08 when the actual proportion is 0.08 (or less). Consequence: The potato chip producer sends the truckload of acceptable potatoes away, which may result in lost revenue for the supplier. A Type II error would occur if the producer does not send the truck away when more than 8% of the potatoes in the shipment have blemishes. Consequence: The producer uses the truckload of potatoes to make potato chips. More chips will be made with blemished potatoes, which may upset consumers. Factors that Increase Power Sample Size The larger the sample size, the higher the power. Alpha Significance Level Increasing alpha (from 0.01 to 0.05) increases the power, because a less conservative alpha increases the chance of (correctly) rejecting the null. Value of the Alternative Parameter The greater the difference between the Hypothesized and True Mean the more obvious the result and therefore the greater the power. More: Increase Power, Decrease Type II The probability of making a Type 1 error is ALWSYS equal to the significance level ( ). The probability of making a Type 1 error is 1. Beta is the power of the test. 6

7 Increase Power, Decrease Type II Blue = Power Red = Type II error Error Probabilities We can assess the performance of a significance test by looking at the probabilities of the two types of error. That s because statistical inference is based on asking, What would happen if I did this many times? For the truckload of potatoes in the previous example, we were testing H 0 : p = 0.08 H a : p > 0.08 where p is the actual proportion of potatoes with blemishes. Suppose that the potato chip producer decides to carry out this test based on a random sample of 500 potatoes using a 5% significance level (α = 0.05). What s Worse? Type I or II It depends. It s impossible to minimize both error types completely. Error Probabilities The shaded area in the right tail is 5%. Sample proportion values to the right of the green line at will cause us to reject H 0 even though H 0 is true. This will happen in 5% of all possible samples. That is, P(making a Type I error) = Type 2 Errors Investigation WS Select: Improved Batting Averages (Power) More on Type 1 and 2 Errors Or direct link: ower/power.html 7

8 Error Probabilities The potato chip producer wonders whether the significance test of H 0 : p = 0.08 versus H a : p > 0.08 based on a random sample of 500 potatoes has enough power to detect a shipment with, say, 11% blemished potatoes. In this case, a particular Type II error is to fail to reject H 0 : p = 0.08 when p = What if p = 0.11? Section 9.1 Significance Tests: The Basics Summary Small P values indicate strong evidence against H 0. To calculate a P value, we must know the sampling distribution of the test statistic when H 0 is true. There is no universal rule for how small a P value in a significance test provides convincing evidence against the null hypothesis. If the P value is smaller than a specified value α (called the significance level), the data are statistically significant at level α. In that case, we can reject H 0. If the P value is greater than or equal to α, we fail to reject H 0. A Type I error occurs if we reject H 0 when it is in fact true. A Type II error occurs if we fail to reject H 0 when it is actually false. In a fixed level α significance test, the probability of a Type I error is the significance level α. The power of a significance test against a specific alternative is the probability that the test will reject H 0 when the alternative is true. Power measures the ability of the test to detect an alternative value of the parameter. For a specific alternative, P(Type II error) = 1 power. Error Probabilities The potato chip producer wonders whether the significance test of H 0 : p = 0.08 versus H a : p > 0.08 based on a random sample of 500 potatoes has enough power to detect a shipment with, say, 11% blemished potatoes. In this case, a particular Type II error is to fail to reject H 0 : p = 0.08 when p = Earlier, we decided to reject H 0 at α = 0.05 if our sample yielded a sample proportion to the right of the green line. Power and Type II Error The power of a test against any alternative is 1 minus the probability of a Type II error for that alternative; that is, power = 1 - β. Since we reject H 0 at α= 0.05 if our sample yields a proportion > , we d correctly reject the shipment about 75% of the time. Section 9.2 Tests About a Population Proportion Section 9.1 Significance Tests: The Basics Summary In this section, we learned that A significance test assesses the evidence provided by data against a null hypothesis H 0 in favor of an alternative hypothesis H a. The hypotheses are stated in terms of population parameters. Often, H 0 is a statement of no change or no difference. H a says that a parameter differs from its null hypothesis value in a specific direction (one sided alternative) or in either direction (two sided alternative). The reasoning of a significance test is as follows. Suppose that the null hypothesis is true. If we repeated our data production many times, would we often get data as inconsistent with H 0 as the data we actually have? If the data are unlikely when H 0 is true, they provide evidence against H 0. The P value of a test is the probability, computed supposing H 0 to be true, that the statistic will take a value at least as extreme as that actually observed in the direction specified by H a. Section 9.2 Tests About a Population Proportion After this section, you should be able to CHECK conditions for carrying out a test about a population proportion. CONDUCT a significance test about a population proportion. CONSTRUCT a confidence interval to draw a conclusion about for a two sided test about a population proportion. 8

9 Recall our basketball player who claimed to be an 80% free throw shooter. In an SRS of 50 freethrows, he made 32. His sample proportion of made shots, 32/50 = 0.64, is much lower than what he claimed. Does it provide convincing evidence against his claim? We use z scores and p values to evaluate. P H A N T O M S Parameters Hypothesis Assess Conditions Name the Test Test Statistic (Calculate) Obtain P value Make a decision State conclusion Theory: One Sample z Test for a Proportion The z statistic has approximately the standard Normal distribution when H 0 is true. Parameters & Hypothesis Parameter: p = the actual proportion of free throws the shooter makes in the long run. Hypothesis: H 0 : p = 0.80 H a : p < 0.80 Theory: Test Statistic and P value A significance test uses sample data to measure the strength of evidence against H 0. The test compares a statistic calculated from sample data with the value of the parameter stated by the null hypothesis. Values of the statistic far from the null parameter value in the direction specified by the alternative hypothesis give evidence against H 0. A test statistic measures how far a sample statistic diverges from what we would expect if the null hypothesis H 0 were true. Assess Conditions: Random, Normal & Independent. Random We can view this set of 50 shots as a simple random sample from the population of all possible shots that the player takes. Normal Assuming H 0 is true, p = then np = (50)(0.80) = 40 and n (1 p) = (50)(0.20) = 10 are both at least 10, so the normal condition is met. Independent In our simulation, the outcome of each shot does is determined by a random number generator, so individual observations are independent. 9

10 Name the Test, Test Statistic (Calculate) & Obtain P value Name Test: One proportion z test Make a Decision & State the Conclusion Make a Decision: P value is Since the p value is so small we reject the null hypothesis. State the Conclusion: We have convincing evidence that the basketball player does not make 80% of his free throws. Z score: 2.83 P value: Name the Test TINspire: Menu, 6: Statistics, 7: Stats Tests, 5: 1 Prop z Test Example: One Potato, Two Potato A potato chip producer has just received a truckload of potatoes from its main supplier. If the producer determines that more than 8% of the potatoes in the shipment have blemishes, the truck will be sent away to get another load from the supplier. A supervisor selects a random sample of 500 potatoes from the truck. An inspection reveals that 47 of the potatoes have blemishes. Carry out a significance test at the α= 0.10 significance level. What should the producer conclude? Test Statistic (Calculate) & Obtain P value Example: One Potato, Two Potato State Parameter & State Hypothesis α = 0.10 significance level Parameter: p = actual proportion of potatoes in this shipment with blemishes. Hypothesis: H 0 : p = 0.08 H a : p >

11 Example: One Potato, Two Potato Assess Check Conditions Random Random sample of 500 potatoes Normal Assuming H 0 : p = 0.08 is true, the expected numbers of blemished and unblemished potatoes are np 0 = 500(0.08) = 40 and n(1 p 0 ) = 500(0.92) = 460, respectively. Because both of these values are at least 10, we should be safe doing Normal calculations. Independent Because we are sampling without replacement, we need to check the 10% condition. It seems reasonable to assume that there are at least 10(500) = 5000 potatoes in the shipment. Example: One Potato, Two Potato Make a Decision and State Conclusion Make a Decision: Since our P value, , is greater than the chosen significance level of α = 0.10, so we fail to reject the null hypothesis. State Conclusion: There is not sufficient evidence to conclude that the shipment contains more than 8% blemished potatoes. The producer will use this truckload of potatoes to make potato chips. Example: One Potato, Two Potato Name the Test Smoking in High School According to the Centers for Disease Control and Prevention (CDC) Web site, 50% of high school students have never smoked a cigarette. Taeyeon wonders whether this national result holds true in his large, urban high school. For his AP Statistics class project, Taeyeon surveys an SRS of 150 students from his school. He gets responses from all 150 students, and 90 say that they have never smoked a cigarette. What should Taeyeon conclude? Give appropriate evidence to support your answer. Example: One Potato, Two Potato Test Statistic (Calculate) and Obtain P value State Parameter & State Hypothesis Perform at test at the α = 0.05 significance level Parameter: p = actual proportion of students in Taeyeon s school who would say they have never smoked cigarettes Hypothesis: H 0 : p = 0.50 H a : p

12 Assess Conditions Random Taeyeon surveyed an SRS of 150 students from his school. Normal Assuming H 0 : p = 0.50 is true, the expected numbers of smokers and nonsmokers in the sample are np 0 = 150(0.50) = 75 and n(1 p 0 ) = 150(0.50) = 75. Because both of these values are at least 10, we should be safe doing Normal calculations. Independent We are sampling without replacement, we need to check the 10% condition. It seems reasonable to assume that there are at least 10(150) = 1500 students a large high school. Confidence Intervals Give More Information The result of a significance test is basically a decision to reject H 0 or fail to reject H 0. When we reject H 0, we re left wondering what the actual proportion p might be. A confidence interval might shed some light on this issue. High School Smoking; our 95% confidence interval is: We are 95% confident that the interval from to captures the true proportion of students at Taeyeon s high school who would say that they have never smoked a cigarette. Name the Test, Test Statistic (Calculate) & Obtain P value Confidence Intervals and Two Sided Tests Name: One Proportion Z Test P0: 0.50 x: 90 n: 150 Test Statistic: z = Obtain p value: p = Make a Decision & State Conclusion Make Decision: Since our P value, , is less than the chosen significance level of α = we have sufficient evidence to reject H 0. Confidence Intervals and Two Sided Tests State Conclusion: We have convincing evidence to conclude that the proportion of students at Taeyeon s school who say they have never smoked differs from the national result of

13 Example: One Potato, Two Potato Step 3: Calculations The sample proportion of blemished potatoes is One Proportion Z Test by Hand P value Using Table A the desired P value is P(z 1.15) = = Basketball Step 3a: Calculate Mean & Standard Deviation If the null hypothesis H 0 : p = 0.80 is true, then the player s sample proportion of made free throws in an SRS of 50 shots would vary according to an approximately Normal sampling distribution with mean High School Smoking The sample proportion is P value To compute this P value, we find the area in one tail and double it. Using Table A or normalcdf(2.45, 100) yields P(z 2.45) = (the righttail area). So the desired P value is2(0.0071) = Basketball Step 3b: Calculate Test Statistic Then, Using Table A, we find that the P value is P(z 2.83) = Section 9.3 Tests About a Population Mean 13

14 Section 9.3 Tests About a Population Mean After this section, you should be able to CHECK conditions for carrying out a test about a population mean. CONDUCT a one sample t test about a population mean. CONSTRUCT a confidence interval to draw a conclusion for a two sided test about a population mean. PERFORM significance tests for paired data. for µ ABC company claimed to have developed a new AAA battery that lasts longer than its regular AAA batteries. Based on years of experience, the company knows that its regular AAA batteries last for 30 hours of continuous use, on average. An SRS of 15 new batteries lasted an average of 33.9 hours with a standard deviation of 9.8 hours. Do these data give convincing evidence that the new batteries last longer on average? Introduction Confidence intervals and significance tests for a population proportion p are based on z values from the standard Normal distribution. Reminder: Inference about a population mean µ uses a t distribution with n 1 degrees of freedom, except in the rare case when the population standard deviation σ is known. State Parameter & State Hypothesis Parameter: µ = the true mean lifetime of the new deluxe AAA batteries. Hypothesis: H 0 : µ = 30 hours H a : µ > 30 hours The One Sample t Test Choose an SRS of size n from a large population that contains an unknown mean µ. To test the hypothesis H 0 : µ = µ 0, compute the one sample t statistic Find the P value by calculating the probability of getting a t statistic this large or larger in the direction specified by the alternative hypothesis H a in a t distribution with df = n 1 Assess Conditions Random, Normal, and Independent. Random: The company tests an SRS of 15 new AAA batteries. Normal: With such a small sample size (n = 15), we need to inspect the data for any departures from Normality. The boxplot show slight rightskewness but no outliers. We should be safe performing a t test about the population mean lifetime µ. Independent Since the batteries are being sampled without replacement, we need to check the 10% condition 10% Condition: There must be at least 10(15) = 150 new AAA batteries. This seems reasonable to believe. 14

15 Name Test, Test statistic (Calculation) and Obtain P value One sample t test t: df = 14 p value: Details: T score table T score table gives a range of possible P values for a significance. We can still draw a conclusion by comparing the range of possible P values to our desired significance level. T score table only includes probabilities only for t distributions with degrees of freedom from 1 to 30 and then skips to df = 40, 50, 60, 80, 100, and (The bottom row gives probabilities for df =, which corresponds to the standard Normal curve.) If the df you need isn t provided in Table B, use the next lower df that is available. T score table shows probabilities only for positive values of t. To find a P value for a negative value of t, we use the symmetry of the t distributions. Make a Decision & State Conclusion Make a Decision: Since the p value of exceeds our α = 0.05 significance level, we fail to reject the null hypothesis and Make a Conclusion: we can t conclude that the company s new AAA batteries last longer than 30 hours, on average. Example: Healthy Streams The level of dissolved oxygen (DO) in a stream or river is an important indicator of the water s ability to support aquatic life. A researcher measures the DO level at 15 randomly chosen locations along a stream. Here are the results in milligrams per liter: A dissolved oxygen level below 5 mg/l puts aquatic life at risk. Details: Normal Condition The Normal condition for means is either population distribution is Normal or sample size is large (n 30). If the sample size is large (n 30), we can safely carry out a significance test (due to the central limit theorem). Example: Healthy Streams State Parameters & State Hypothesis α = 0.05 Parameters: µ = actual mean dissolved oxygen level in this stream. If the sample size is small, we should examine (create graph on calculator and then DRAW on paper) the sample data for any obvious departures from Normality, such as skewness and outliers. Hypothesis: H 0 : µ = 5 H a : µ < 5 15

16 Example: Healthy Streams Assess Conditions Random The researcher measured the DO level at 15 randomly chosen locations. Normal With such a small sample size (n = 15), we need to look at (and DRAW) the data. The boxplot shows no outliers; with no outliers or strong skewness, therefore we can use t procedures. Example: Healthy Streams Make a Decision & State Conclusion Make Decision: Since the P value is and this is greater than our α = 0.05 significance level, we fail to reject H 0. State Conclusion: We don t have enough evidence to conclude that the mean DO level in the stream is less than 5 mg/l. Independent There is an infinite number of possible locations along the stream, so it isn t necessary to check the 10% condition. We do need to assume that individual measurements are independent. Example: Healthy Streams Name Test Name Test: One Sample T Test Pineapples: Two Sided Tests At the Hawaii Pineapple Company, managers are interested in the sizes of the pineapples grown in the company s fields. Last year, the mean weight of the pineapples harvested from one large field was 31 ounces. A new irrigation system was installed in this field after the growing season. Managers wonder whether this change will affect the mean weight of future pineapples grown in the field. To find out, they select and weigh a random sample of 50 pineapples from this year s crop. The Minitab output below summarizes the data. ***Enter data into list first and name stream Example: Healthy Streams Test Statistic (Calculate) and Obtain P value State Parameters & State Hypothesis: Parameters: µ = the mean weight (in ounces) of all pineapples grown in the field this year Hypothesis: H 0 : µ = 31 H a : µ 31 Since no significance level is given, we ll use α = Test Statistic: t=.9426 with df= 14 P value:

17 Assess Conditions Random The data came from a random sample of 50 pineapples from this year s crop. Normal We don t know whether the population distribution of pineapple weights this year is Normally distributed. But n = 50 30, so the large sample size (and the fact that there are no outliers) makes it OK to use t procedures. Independent There need to be at least 10(50) = 500 pineapples in the field because managers are sampling without replacement (10% condition). We would expect many more than 500 pineapples in a large field. Confidence Intervals Give More Information Minitab output for a significance test and confidence interval based on the pineapple data is shown below. The test statistic and P value match what we got earlier (up to rounding). The 95% confidence interval for the mean weight of all the pineapples grown in the field this year is to ounces. We are 95% confident that this interval captures the true mean weight µ of this year s pineapple crop. Name Test, Test Statistic (Calculate) and Obtain P value Confidence Intervals and Two Sided Tests The connection between two sided tests and confidence intervals is even stronger for means than it was for proportions. That s because both inference methods for means use the standard error of the sample mean in the calculations. A two sided test at significance level α (say, α = 0.05) and a 100(1 α)% confidence interval (a 95% confidence interval if α = 0.05) give similar information about the population parameter. When the two sided significance test at level α rejects H 0 : µ = µ 0, the 100(1 α)% confidence interval for µ will not contain the hypothesized value µ 0. When the two sided significance test at level α fails to reject the null hypothesis, the confidence interval for µ will contain µ 0. Make Decision & State Conclusion Make Decision: Since the P value is it is less than our α = 0.05 significance level, so we have enough evidence to reject the null hypothesis. State Conclusion: We have convincing evidence that the mean weight of the pineapples in this year s crop is not 31 ounces; meaning the irrigation system has an effect.* * We only KNOW that there is an effect. We did not test whether the effect was positive (bigger pineapples) or negative. Inference for Means: Paired Data Study designs that involve making two observations on the same individual, or one observation on each of two similar individuals, result in paired data. When paired data result from measuring the same quantitative variable twice, we can make comparisons by analyzing the differences in each pair. If the conditions for inference are met, we can use onesample t procedures to perform inference about the mean difference µ d. These methods are sometimes called paired t procedures. 17

18 Caffeine: Paired Data Researchers designed an experiment to study the effects of caffeine withdrawal. They recruited 11 volunteers who were diagnosed as being caffeine dependent to serve as subjects. Each subject was barred from coffee, colas, and other substances with caffeine for the duration of the experiment. During one twoday period, subjects took capsules containing their normal caffeine intake. During another two day period, they took placebo capsules. The order in which subjects took caffeine and the placebo was randomized. At the end of each two day period, a test for depression was given to all 11 subjects. Researchers wanted to know whether being deprived of caffeine would lead to an increase in depression. A higher value equals higher levels of depression. Data on next slide. Assess Conditions: Random researchers randomly assigned the treatment order placebo then caffeine, caffeine then placebo to the subjects. Normal We don t know whether the actual distribution of difference in depression scores (placebo caffeine) is Normal. So, with such a small sample size (n = 11), we need to examine the data The boxplot shows some rightskewness but no outliers; with no outliers or strong skewness, the t procedures are reasonable to use. Independent We aren t sampling, so it isn t necessary to check the 10% condition. We will assume that the changes in depression scores for individual subjects are independent. This is reasonable if the experiment is conducted properly. Subject Results of a caffeine deprivation study Depression (caffeine) Depression (placebo) Difference (placebo caffeine) Name Test, Test Statistic (Calculate) and Obtain P value State Parameters & State Hypothesis: If caffeine deprivation has no effect on depression, then we would expect the actual mean difference in depression scores to be 0. Parameter: µ d = the true mean difference (placebo caffeine) in depression score. Hypotheses: H 0 : µ d = 0 H a : µ d > 0 Make Decision & State Conclusion: Make Decision: Since the P value of is much less than our chosen α = 0.05, we have convincing evidence to reject H 0 : µ d = 0. State Conclusion: We can therefore conclude that depriving these caffeine dependent subjects of caffeine caused* an average increase in depression scores. *Since the data came from a well designed experiment we can use the word caused Since no significance level is given, we ll use α =

19 Using Tests Wisely: Statistical Significance and Practical Importance Statistical significance is valued because it points to an effect that is unlikely to occur simply by chance. When a null hypothesis ( no effect or no difference ) can be rejected at the usual levels (α = 0.05 or α = 0.01), there is good evidence of a difference. But that difference may be very small. When large samples are available, even tiny deviations from the null hypothesis will be significant. T Tests by Hand Using Tests Wisely: Don t Ignore Lack of Significance There is a tendency to infer that there is no difference whenever a P value fails to attain the usual 5% standard. In some areas of research, small differences that are detectable only with large sample sizes can be of great practical significance. When planning a study, verify that the test you plan to use has a high probability (power) of detecting a difference of the size you hope to find. Example: Healthy Streams Step 3: Calculations The sample mean and standard deviation are Upper-tail probability p df % 60% 70% Confidence level C P value The P value is the area to the left of t = 0.94 under the t distribution curve with df = 15 1 = 14. Using Tests Wisely: Statistical Inference Is Not Valid for All Sets of Data Pineapples The sample mean and standard deviation are Badly designed surveys or experiments often produce invalid results. Formal statistical inference cannot correct basic flaws in the design. Each test is valid only in certain circumstances, with properly produced data being particularly important. Crap in = crap out Upper-tail probability p df % 99.5% 99.8% Confidence level C P value The P value for this two sided test is the area under the t distribution curve with 50 1 = 49 degrees of freedom. Since Table B does not have an entry for df = 49, we use the more conservative df = 40. The upper tail probability is between and so the desired P value is between 0.01 and

20 Caffeine Calculations The sample mean and standard deviation are Section 9.3 Tests About a Population Mean Summary In this section, we learned that Significance tests for the mean µ of a Normal population are based on the sampling distribution of the sample mean. Due to the central limit theorem, the resulting procedures are approximately correct for other population distributions when the sample is large. P value According to technology, the area to the right of t = 3.53 on the t distribution curve with df = 11 1 = 10 is If we somehow know σ, we can use a z test statistic and the standard Normal distribution to perform calculations. In practice, we typically do not know σ. Then, we use the one sample t statistic with P values calculated from the t distribution with n 1 degrees of freedom. Name Test, Test statistic (Calculation) and Obtain P value For a test of H 0 : µ = µ 0, our statistic is the sample mean. Its standard deviation is Because the population standard deviation σ is usually unknown, we use the sample standard deviation s x in its place. The resulting test statistic has the standard error of the sample mean in the denominator When the Normal condition is met, this statistic has a t distribution with n 1 degrees of freedom. Summary Section 9.3 Tests About a Population Mean The one sample t test is approximately correct when Random The data were produced by random sampling or a randomized experiment. Normal The population distribution is Normal OR the sample size is large (n 30). Independent Individual observations are independent. When sampling without replacement, check that the population is at least 10 times as large as the sample. Confidence intervals provide additional information that significance tests do not namely, a range of plausible values for the parameter µ. A twosided test of H 0 : µ = µ 0 at significance level α gives the same conclusion as a 100(1 α)% confidence interval for µ. Analyze paired data by first taking the difference within each pair to produce a single sample. Then use one sample t procedures. Carrying Out a Hypothesis Test Name Test, Test statistic (Calculation) and Obtain P value Summary Section 9.3 Tests About a Population Mean Upper-tail probability p df % 90% 95% Confidence level C The P value is the probability of getting a result this large or larger in the direction indicated by H a, that is, P(t 1.54). Go to the df = 14 row. Since the t statistic falls between the values and 1.761, the Upper tail probability p is between 0.10 and The P value for this test is between 0.05 and Very small differences can be highly significant (small P value) when a test is based on a large sample. A statistically significant difference need not be practically important. Lack of significance does not imply that H 0 is true. Even a large difference can fail to be significant when a test is based on a small sample. Significance tests are not always valid. Faulty data collection, outliers in the data, and other practical problems can invalidate a test. Many tests run at once will probably produce some significant results by chance alone, even if all the null hypotheses are true. 20