Outcomes and the Type I and Type II Errors When you perform a hypothesis test, there are four possible outcomes depending on the actual truth (or falseness) of the null hypothesis H 0 and the decision to reject or not. The outcomes are summarized in the following table: ACTION H 0 IS ACTUALLY ... True False Do not reject H 0 Correct Outcome Type II error Reject H 0 Type I Error Correct Outcome The four possible outcomes in the table are: The decision is not to reject H 0 when H 0 is true (correct decision). The decision is to reject H 0 when H 0 is true (incorrect decision known as a Type I error ). The decision is not to reject H 0 when, in fact, H 0 is false (incorrect decision known as a Type II error ). The decision is to reject H 0 when H 0 is false ( correct decision whose probability is called the Power of the Test ). Each of the errors occurs with a particular probability. The Greek letters α and β represent the probabilities. α = probability of a Type I error = P (Type I error) = probability of rejecting the null hypothesis when the null hypothesis is true. β = probability of a Type II error = P (Type II error) = probability of not rejecting the null hypothesis when the null hypothesis is false. α and β should be as small as possible because they are probabilities of errors. They are rarely zero. The Power of the Test is 1 – β . Ideally, we want a high power that is as close to one as possible. Increasing the sample size can increase the Power of the Test. The following are examples of Type I and Type II errors. Suppose the null hypothesis, H 0 , is: Navah's rock climbing equipment is safe. Type I error : Navah thinks that her rock climbing equipment may not be safe when, in fact, it really is safe. Type II error : Navah thinks that her rock climbing equipment may be safe when, in fact, it is not safe. α = probability that Navah thinks her rock climbing equipment may not be safe when, in fact, it really is safe. β = probability that Navah thinks her rock climbing equipment may be safe when, in fact, it is not safe. Notice that, in this case, the error with the greater consequence is the Type II error. (If Navah thinks her rock climbing equipment is safe, she will go ahead and use it.) Try It Suppose the null hypothesis, H 0 , is: the blood cultures contain no traces of pathogen X . State the Type I and Type II errors. Suppose the null hypothesis, H 0 , is: The victim of an automobile accident is alive when they arrive at the emergency room of a hospital. Type I error : The emergency crew thinks that the victim is dead when, in fact, the victim is alive. Type II error : The emergency crew does not know if the victim is alive when, in fact, the victim is dead. α = probability that the emergency crew thinks the victim is dead when, in fact, the victim is really alive = P (Type I error). β = probability that the emergency crew does not know if the victim is alive when, in fact, the victim is dead = P (Type II error). The error with the greater consequence is the Type I error. (If the emergency crew thinks the victim is dead, they will not treat them.) Try It Suppose the null hypothesis, H 0 , is: a patient is not sick. Which type of error has the greater consequence, Type I or Type II? A company called Genetic Labs claims to be able to increase the likelihood that a pregnancy will result in a male being born. Statisticians want to test the claim. Suppose that the null hypothesis, H 0 , is: Genetic Labs has no effect on sex outcome. Type I error : This results when a true null hypothesis is rejected. In the context of this scenario, we would state that we believe that Genetic Labs influences the sex outcome, when in fact it has no effect. The probability of this error occurring is denoted by the Greek letter alpha, α . Type II error : This results when we fail to reject a false null hypothesis. In context, we would state that Genetic Labs does not influence the sex outcome of a pregnancy when, in fact, it does. The probability of this error occurring is denoted by the Greek letter beta, β . The error of greater consequence would be the Type I error since people would use the Genetic Labs product in hopes of increasing the chances of having a male. Try It “Red tide” is a bloom of poison-producing algae–a few different species of a class of plankton called dinoflagellates. When the weather and water conditions cause these blooms, shellfish such as clams living in the area develop dangerous levels of a paralysis-inducing toxin. In Massachusetts, the Division of Marine Fisheries (DMF) monitors levels of the toxin in shellfish by regular sampling of shellfish along the coastline. If the mean level of toxin in clams exceeds 800 μg (micrograms) of toxin per kg of clam meat in any area, clam harvesting is banned there until the bloom is over and levels of toxin in clams subside. Describe both a Type I and a Type II error in this context, and state which error has the greater consequence. A certain experimental drug claims a cure rate of at least 75% for males with prostate cancer. Describe both the Type I and Type II errors in context. Which error is the more serious? Type I : A cancer patient believes the cure rate for the drug is less than 75% when it actually is at least 75%. Type II : A cancer patient believes the experimental drug has at least a 75% cure rate when it has a cure rate that is less than 75%. In this scenario, the Type II error contains the more severe consequence. If a patient believes the drug works at least 75% of the time, this most likely will influence the patient’s (and doctor’s) choice about whether to use the drug as a treatment option. Try It Determine both Type I and Type II errors for the following scenario: Assume a null hypothesis, H 0 , that states the percentage of adults with jobs is at least 88%. Identify the Type I and Type II errors from these four statements. Not to reject the null hypothesis that the percentage of adults who have jobs is at least 88% when that percentage is actually less than 88% Not to reject the null hypothesis that the percentage of adults who have jobs is at least 88% when the percentage is actually at least 88%. Reject the null hypothesis that the percentage of adults who have jobs is at least 88% when the percentage is actually at least 88%. Reject the null hypothesis that the percentage of adults who have jobs is at least 88% when that percentage is actually less than 88%. Chapter Review In every hypothesis test, the outcomes are dependent on a correct interpretation of the data. Incorrect calculations or misunderstood summary statistics can yield errors that affect the results. A Type I error occurs when a true null hypothesis is rejected. A Type II error occurs when a false null hypothesis is not rejected. The probabilities of these errors are denoted by the Greek letters α and β , for a Type I and a Type II error respectively. The power of the test, 1 – β , quantifies the likelihood that a test will yield the correct result of a true alternative hypothesis being accepted. A high power is desirable. Formula Review α = probability of a Type I error = P (Type I error) = probability of rejecting the null hypothesis when the null hypothesis is true. β = probability of a Type II error = P (Type II error) = probability of not rejecting the null hypothesis when the null hypothesis is false. The mean price of mid-sized cars in a region is $32,000. A test is conducted to see if the claim is true. State the Type I and Type II errors in complete sentences. Type I: The mean price of mid-sized cars is $32,000, but we conclude that it is not $32,000. Type II: The mean price of mid-sized cars is not $32,000, but we conclude that it is $32,000. A sleeping bag is tested to withstand temperatures of –15 °F. You think the bag cannot stand temperatures that low. State the Type I and Type II errors in complete sentences. For Exercise 9.12 , what are α and β in words? α = the probability that you think the bag cannot withstand -15 degrees F, when in fact it can β = the probability that you think the bag can withstand -15 degrees F, when in fact it cannot In words, describe 1 – β For Exercise 9.12 . A group of doctors is deciding whether or not to perform an operation. Suppose the null hypothesis, H 0 , is: the surgical procedure will go well. State the Type I and Type II errors in complete sentences. Type I: The procedure will go well, but the doctors think it will not. Type II: The procedure will not go well, but the doctors think it will. A group of doctors is deciding whether or not to perform an operation. Suppose the null hypothesis, H 0 , is: the surgical procedure will go well. Which is the error with the greater consequence? The power of a test is 0.981. What is the probability of a Type II error? 0.019 A group of divers is exploring an old sunken ship. Suppose the null hypothesis, H 0 , is: the sunken ship does not contain buried treasure. State the Type I and Type II errors in complete sentences. A microbiologist is testing a water sample for E-coli. Suppose the null hypothesis, H 0 , is: the sample does not contain E-coli. The probability that the sample does not contain E-coli, but the microbiologist thinks it does is 0.012. The probability that the sample does contain E-coli, but the microbiologist thinks it does not is 0.002. What is the power of this test? 0.998 A microbiologist is testing a water sample for E-coli. Suppose the null hypothesis, H 0 , is: the sample contains E-coli. Which is the error with the greater consequence? Homework State the Type I and Type II errors in complete sentences given the following statements. The mean number of years Americans work before retiring is 34. At most 60% of Americans vote in presidential elections. The mean starting salary for San Jose State University graduates is at least $100,000 per year. Twenty-nine percent of high school seniors get drunk each month. Fewer than 5% of adults ride the bus to work in Los Angeles. The mean number of cars a person owns in their lifetime is not more than ten. About half of Americans prefer to live away from cities, given the choice. Europeans have a mean paid vacation each year of six weeks. The chance of developing breast cancer is under 11% for females. Private universities mean tuition cost is more than $20,000 per year. Type I error: We conclude that the mean is not 34 years, when it really is 34 years. Type II error: We conclude that the mean is 34 years, when in fact it really is not 34 years. Type I error: We conclude that more than 60% of Americans vote in presidential elections, when the actual percentage is at most 60%.Type II error: We conclude that at most 60% of Americans vote in presidential elections when, in fact, more than 60% do. Type I error: We conclude that the mean starting salary is less than $100,000, when it really is at least $100,000. Type II error: We conclude that the mean starting salary is at least $100,000 when, in fact, it is less than $100,000. Type I error: We conclude that the proportion of high school seniors who get drunk each month is not 29%, when it really is 29%. Type II error: We conclude that the proportion of high school seniors who get drunk each month is 29% when, in fact, it is not 29%. Type I error: We conclude that fewer than 5% of adults ride the bus to work in Los Angeles, when the percentage that do is really 5% or more. Type II error: We conclude that 5% or more adults ride the bus to work in Los Angeles when, in fact, fewer that 5% do. Type I error: We conclude that the mean number of cars a person owns in their lifetime is more than 10, when in reality it is not more than 10. Type II error: We conclude that the mean number of cars a person owns in their lifetime is not more than 10 when, in fact, it is more than 10. Type I error: We conclude that the proportion of Americans who prefer to live away from cities is not about half, though the actual proportion is about half. Type II error: We conclude that the proportion of Americans who prefer to live away from cities is half when, in fact, it is not half. Type I error: We conclude that the duration of paid vacations each year for Europeans is not six weeks, when in fact it is six weeks. Type II error: We conclude that the duration of paid vacations each year for Europeans is six weeks when, in fact, it is not. Type I error: We conclude that the proportion is less than 11%, when it is really at least 11%. Type II error: We conclude that the proportion of females who develop breast cancer is at least 11%, when in fact it is less than 11%. Type I error: We conclude that the average tuition cost at private universities is more than $20,000, though in reality it is at most $20,000. Type II error: We conclude that the average tuition cost at private universities is at most $20,000 when, in fact, it is more than $20,000. For statements a-j in Exercise 9.66 , answer the following in complete sentences. State a consequence of committing a Type I error. State a consequence of committing a Type II error. When a new drug is created, the pharmaceutical company must subject it to testing before receiving the necessary permission from the Food and Drug Administration (FDA) to market the drug. Suppose the null hypothesis is “the drug is unsafe.” What is the Type II Error? To conclude the drug is safe when in, fact, it is unsafe. Not to conclude the drug is safe when, in fact, it is safe. To conclude the drug is safe when, in fact, it is safe. Not to conclude the drug is unsafe when, in fact, it is unsafe. b A statistics instructor believes that fewer than 20% of Evergreen Valley College (EVC) students attended the opening midnight showing of the latest Marvel Universe movie. The instructor surveys 84 students and finds that 11 of them attended the midnight showing. The Type I error is to conclude that the percent of EVC students who attended is ________. at least 20%, when in fact, it is less than 20%. 20%, when in fact, it is 20%. less than 20%, when in fact, it is at least 20%. less than 20%, when in fact, it is less than 20%. It is believed that Lake Tahoe Community College (LTCC) Intermediate Algebra students get less than seven hours of sleep per night, on average. A survey of 22 LTCC Intermediate Algebra students generated a mean of 7.24 hours with a standard deviation of 1.93 hours. At a level of significance of 5%, do LTCC Intermediate Algebra students get less than seven hours of sleep per night, on average? The Type II error is not to reject that the mean number of hours of sleep LTCC students get per night is at least seven when, in fact, the mean number of hours is more than seven hours. is at most seven hours. is at least seven hours. is less than seven hours. d Previously, an organization reported that teenagers spent 4.5 hours per week, on average, on the phone. The organization thinks that, currently, the mean is higher. Fifteen randomly chosen teenagers were asked how many hours per week they spend on the phone. The sample mean was 4.75 hours with a sample standard deviation of 2.0. Conduct a hypothesis test, the Type I error is: to conclude that the current mean hours per week is higher than 4.5, when in fact, it is higher to conclude that the current mean hours per week is higher than 4.5, when in fact, it is the same to conclude that the mean hours per week currently is 4.5, when in fact, it is higher to conclude that the mean hours per week currently is no higher than 4.5, when in fact, it is not higher Type I Error The decision is to reject the null hypothesis when, in fact, the null hypothesis is true. Type II Error The decision is not to reject the null hypothesis when, in fact, the null hypothesis is false.

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Probability Distribution Needed for Hypothesis Testing Earlier in the course, we discussed sampling distributions. Particular distributions are associated with various types of hypothesis testing. The following table summarizes various hypothesis tests and corresponding probability distributions that will be used to conduct the test (based on the assumptions shown below): Type of Hypothesis Test Population Parameter Estimated value (point estimate) Probability Distribution Used Hypothesis test for the mean, when the population standard deviation is known Population mean μ Sample mean x ¯ Normal distribution, X ¯ ~ N ( μ X , σ X n ) Hypothesis test for the mean, when the population standard deviation is unknown and the distribution of the sample mean is approximately normal Population mean μ Sample mean x ¯ Student’s t-distribution, t d f Hypothesis test for proportions Population proportion p Sample proportion p ' Normal distribution, P ' ~ N ( p , p · q n ) Assumptions When you perform a hypothesis test of a single population mean μ using a normal distribution (often called a z-test), you take a simple random sample from the population. The population you are testing is normally distributed , or your sample size is sufficiently large. You know the value of the population standard deviation , which, in reality, is rarely known. When you perform a hypothesis test of a single population mean μ using a Student's t-distribution (often called a t -test), there are fundamental assumptions that need to be met in order for the test to work properly. Your data should be a simple random sample that comes from a population that is approximately normally distributed. You use the sample standard deviation to approximate the population standard deviation. (Note that if the sample size is sufficiently large, a t -test will work even if the population is not approximately normally distributed). When you perform a hypothesis test of a single population proportion p , you take a simple random sample from the population. You must meet the conditions for a binomial distribution : there are a certain number n of independent trials, the outcomes of any trial are success or failure, and each trial has the same probability of a success p . The shape of the binomial distribution needs to be similar to the shape of the normal distribution. To ensure this, the quantities np and nq must both be greater than five ( n p > 5 and n q > 5 ). Then the binomial distribution of a sample (estimated) proportion can be approximated by the normal distribution with μ = p and σ = p q n . Remember that q = 1 - p . Chapter Review In order for a hypothesis test’s results to be generalized to a population, certain requirements must be satisfied. When testing for a single population mean: A Student's t -test should be used if the data come from a simple, random sample and the population is approximately normally distributed, or the sample size is large, with an unknown standard deviation. The normal test will work if the data come from a simple, random sample and the population is approximately normally distributed, or the sample size is large, with a known standard deviation. When testing a single population proportion use a normal test for a single population proportion if the data comes from a simple, random sample, fill the requirements for a binomial distribution, and the mean number of successes and the mean number of failures satisfy the conditions: np > 5 and nq > 5 where n is the sample size, p is the probability of a success, and q is the probability of a failure. Formula Review If there is no given preconceived α , then use α = 0.05. Types of Hypothesis Tests Single population mean, known population variance (or standard deviation): Normal test . Single population mean, unknown population variance (or standard deviation): Student's t -test . Single population proportion: Normal test . For a single population mean , we may use a normal distribution with the following mean and standard deviation. Means: μ = μ x ¯ and σ x ¯ = σ x n A single population proportion , we may use a normal distribution with the following mean and standard deviation. Proportions: µ = p and σ = p q n . Which two distributions can you use for hypothesis testing for this chapter? A normal distribution or a Student’s t -distribution Which distribution do you use when you are testing a population mean and the population standard deviation is known? Assume a normal distribution, with n ≥ 30. Which distribution do you use when the standard deviation is not known and you are testing one population mean? Assume sample size is large. Use a Student’s t -distribution A population mean is 13. The sample mean is 12.8, and the sample standard deviation is two. The sample size is 20. What distribution should you use to perform a hypothesis test? Assume the underlying population is normal. A population has a mean of 25 and a standard deviation of five. The sample mean is 24, and the sample size is 108. What distribution should you use to perform a hypothesis test? a normal distribution for a single population mean It is thought that 42% of respondents in a taste test would prefer Brand A . In a particular test of 100 people, 39% preferred Brand A . What distribution should you use to perform a hypothesis test? You are performing a hypothesis test of a single population mean using a Student’s t -distribution. What must you assume about the distribution of the data? It must be approximately normally distributed. You are performing a hypothesis test of a single population mean using a Student’s t -distribution. The data are not from a simple random sample. Can you accurately perform the hypothesis test? You are performing a hypothesis test of a single population proportion. What must be true about the quantities of np and nq ? They must both be greater than five. You are performing a hypothesis test of a single population proportion. You find out that np is less than five. What must you do to be able to perform a valid hypothesis test? You are performing a hypothesis test of a single population proportion. The data come from which distribution? binomial distribution Homework It is believed that Lake Tahoe Community College (LTCC) Intermediate Algebra students get less than seven hours of sleep per night, on average. A survey of 22 LTCC Intermediate Algebra students generated a mean of 7.24 hours with a standard deviation of 1.93 hours. At a level of significance of 5%, do LTCC Intermediate Algebra students get less than seven hours of sleep per night, on average? The distribution to be used for this test is X ¯ ~ ________________ N ( 7.24 , 1.93 22 ) N ( 7.24 , 1.93 ) t 22 t 21 d Binomial Distribution a discrete random variable (RV) that arises from Bernoulli trials. There are a fixed number, n , of independent trials. “Independent” means that the result of any trial (for example, trial 1) does not affect the results of the following trials, and all trials are conducted under the same conditions. Under these circumstances the binomial RV Χ is defined as the number of successes in n trials. The notation is: X ~ B(n, p) μ = np and the standard deviation is σ = n p q . The probability of exactly x successes in n trials is P ( X = x ) = ( n x ) p x q n − x . Normal Distribution a continuous random variable (RV) with pdf f ( x ) = 1 σ 2 π e − ( x − μ ) 2 2 σ 2 , where μ is the mean of the distribution, and σ is the standard deviation, notation: X ~ N ( μ , σ ). If μ = 0 and σ = 1, the RV is called the standard normal distribution . Standard Deviation a number that is equal to the square root of the variance and measures how far data values are from their mean; notation: s for sample standard deviation and σ for population standard deviation. Student's t -Distribution investigated and reported by William S. Gossett in 1908 and published under the pseudonym Student. The major characteristics of the random variable (RV) are: It is continuous and assumes any real values. The pdf is symmetrical about its mean of zero. However, it is more spread out and flatter at the apex than the normal distribution. It approaches the standard normal distribution as n gets larger. There is a "family" of t distributions: every representative of the family is completely defined by the number of degrees of freedom which is one less than the number of data items.

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Rare Events, the Sample, Decision and Conclusion Establishing the type of distribution, sample size, and known or unknown standard deviation can help you figure out how to go about a hypothesis test. However, there are several other factors you should consider when working out a hypothesis test. Rare Events Suppose you make an assumption about a property of the population (this assumption is the null hypothesis ). Then you gather sample data randomly. If the sample has properties that would be very unlikely to occur if the assumption is true, then you would conclude that your assumption about the population is probably incorrect. (Remember that your assumption is just an assumption —it is not a fact and it may or may not be true. But your sample data are real and the data are showing you a fact that seems to contradict your assumption.) For example, Didi and Ali are at a birthday party of a very wealthy friend. They hurry to be first in line to grab a prize from a tall basket that they cannot see inside because they will be blindfolded. There are 200 plastic bubbles in the basket and Didi and Ali have been told that there is only one with a $100 bill. Didi is the first person to reach into the basket and pull out a bubble. Her bubble contains a $100 bill. The probability of this happening is 1 200 = 0.005. Because this is so unlikely, Ali is hoping that what the two of them were told is wrong and there are more $100 bills in the basket. A "rare event" has occurred (Didi getting the $100 bill) so Ali doubts the assumption about only one $100 bill being in the basket. Using the Sample to Test the Null Hypothesis Use the sample data to calculate the actual probability of getting the test result, called the p -value . The p -value is the probability that, if the null hypothesis is true, the results from another randomly selected sample will be as extreme or more extreme as the results obtained from the given sample. A large p -value calculated from the data indicates that we should not reject the null hypothesis . The smaller the p -value, the more unlikely the outcome, and the stronger the evidence is against the null hypothesis. We would reject the null hypothesis if the evidence is strongly against it. Draw a graph that shows the p -value. The hypothesis test is easier to perform if you use a graph because you see the problem more clearly. Suppose a baker claims that his bread height is more than 15 cm, on average. Several of his customers do not believe him. To persuade his customers that he is right, the baker decides to do a hypothesis test. He bakes 10 loaves of bread. The mean height of the sample loaves is 17 cm. The baker knows from baking hundreds of loaves of bread that the standard deviation for the height is 0.5 cm. and the distribution of heights is normal. The null hypothesis could be H 0 : μ ≤ 15 The alternate hypothesis is H a : μ > 15 The words "is more than" translates as a ">" so " μ > 15" goes into the alternate hypothesis. The null hypothesis must contradict the alternate hypothesis. By the Central Limit Theorem , the distribution of sample means, for samples of 10 loaves, will be normal with mean μ = 15 and standard deviation σ n = 0.5 10 = 0.16 . Suppose the null hypothesis is true (the mean height of the loaves is no more than 15 cm). Then is the mean height (17 cm) calculated from the sample unexpectedly large? The hypothesis test works by asking the question how unlikely the sample mean would be if the null hypothesis were true. The graph shows how far out the sample mean is on the normal curve. The p -value is the probability that, if we were to take other samples, any other sample mean would fall at least as far out as 17 cm. The p -value, then, is the probability that a sample mean is the same or greater than 17 cm. when the population mean is, in fact, 15 cm. We can calculate this probability using the normal distribution for means. p -value= P ( x ¯ > 17) which is approximately zero. A p -value of approximately zero tells us that it is highly unlikely that a loaf of bread rises no more than 15 cm, on average. That is, almost 0% of all loaves of bread would be at least as high as 17 cm. purely by CHANCE had the population mean height really been 15 cm. Because the outcome of 17 cm. is so unlikely (meaning it is happening NOT by chance alone) , we conclude that the evidence is strongly against the null hypothesis (the mean height is at most 15 cm.). There is sufficient evidence that the true mean height for the population of the baker's loaves of bread is greater than 15 cm. Try It A normal distribution has a standard deviation of 1. We want to verify a claim that the mean is greater than 12. A sample of 36 is taken with a sample mean of 12.5. H 0 : μ ≤ 12 H a : μ > 12 The p -value is 0.0013 Draw a graph that shows the p -value. Decision and Conclusion A systematic way to make a decision of whether to reject or not reject the null hypothesis is to compare the p -value and a preset or preconceived α (also called a "significance level") . A preset α is the probability of a Type I error (rejecting the null hypothesis when the null hypothesis is true). It may or may not be given to you at the beginning of the problem. When you make a decision to reject or not reject H 0 , do as follows: If α > p -value, reject H 0 . The results of the sample data are significant. There is sufficient evidence to conclude that H 0 is an incorrect belief and that the alternative hypothesis , H a , may be correct. If α ≤ p -value, do not reject H 0 . The results of the sample data are not significant.There is not sufficient evidence to conclude that the alternative hypothesis, H a , may be correct. When you "do not reject H 0 ", it does not mean that you should believe that H 0 is true. It simply means that the sample data have failed to provide sufficient evidence to cast serious doubt about the truthfulness of H o . Conclusion: After you make your decision, write a thoughtful conclusion about the hypotheses in terms of the given problem. When using the p -value to evaluate a hypothesis test, it is sometimes useful to use the following memory device If the p -value is low, the null must go. If the p -value is high, the null must fly. This memory aid relates a p -value less than the established alpha (the p is low) as rejecting the null hypothesis and, likewise, relates a p -value higher than the established alpha (the p is high) as not rejecting the null hypothesis. Fill in the blanks. Reject the null hypothesis when ______________________________________. The results of the sample data _____________________________________. Do not reject the null when hypothesis when __________________________________________. The results of the sample data ____________________________________________. Reject the null hypothesis when the p -value is less than the established alpha value . The results of the sample data support the alternative hypothesis . Do not reject the null hypothesis when the p -value is greater than the established alpha value . The results of the sample data do not support the alternative hypothesis . Try It It’s a Boy Genetics Labs claim their procedures improve the chances of a boy being born. The results for a test of a single population proportion are as follows: H 0 : p = 0.50, H a : p > 0.50 α = 0.01 p -value = 0.025 Interpret the results and state a conclusion in simple, non-technical terms. Chapter Review When the probability of an event occurring is low, and it happens, it is called a rare event. Rare events are important to consider in hypothesis testing because they can inform your willingness not to reject or to reject a null hypothesis. To test a null hypothesis, find the p -value for the sample data and graph the results. When deciding whether or not to reject the null the hypothesis, keep these two parameters in mind: α > p -value, reject the null hypothesis α ≤ p -value, do not reject the null hypothesis When do you reject the null hypothesis? The probability of winning the grand prize at a particular carnival game is 0.005. Is the outcome of winning very likely or very unlikely? The outcome of winning is very unlikely. The probability of winning the grand prize at a particular carnival game is 0.005. Michele wins the grand prize. Is this considered a rare or common event? Why? It is believed that the mean height of high school students who play basketball on the school team is 73 inches with a standard deviation of 1.8 inches. A random sample of 40 players is chosen. The sample mean was 71 inches, and the sample standard deviation was 1.5 inches. Do the data support the claim that the mean height is less than 73 inches? The p -value is almost zero. State the null and alternative hypotheses and interpret the p -value. H 0 : μ > = 73 H a : μ < 73 The p -value is almost zero, which means there is sufficient data to conclude that the mean height of high school students who play basketball on the school team is less than 73 inches at the 5% level. The data do support the claim. The mean age of graduate students at a University is at most 31 y ears with a standard deviation of two years. A random sample of 15 graduate students is taken. The sample mean is 32 years and the sample standard deviation is three years. Are the data significant at the 1% level? The p -value is 0.0264. State the null and alternative hypotheses and interpret the p -value. Does the shaded region represent a low or a high p -value compared to a level of significance of 1%? The shaded region shows a low p -value. What should you do when α > p -value? What should you do if α = p -value? Do not reject H 0 . If you do not reject the null hypothesis, then it must be true. Is this statement correct? State why or why not in complete sentences. Use the following information to answer the next seven exercises: Suppose that a recent article stated that the mean time spent in jail by a first-time convicted burglar is 2.5 years. A study was then done to see if the mean time has increased in the new century. A random sample of 26 first-time convicted burglars in a recent year was picked. The mean length of time in jail from the survey was three years with a standard deviation of 1.8 years. Suppose that it is somehow known that the population standard deviation is 1.5. Conduct a hypothesis test to determine if the mean length of jail time has increased. Assume the distribution of the jail times is approximately normal. Is this a test of means or proportions? means What symbol represents the random variable for this test? In words, define the random variable for this test. the mean time spent in jail for 26 first time convicted burglars Is σ known and, if so, what is it? Calculate the following: x ¯ _______ σ _______ s x _______ n _______ 3 1.5 1.8 26 Since both σ and s x are given, which should be used? In one to two complete sentences, explain why. State the distribution to use for the hypothesis test. X ¯ ~ N ( 2.5 , 1.5 26 ) A random survey of 75 student loan recipients revealed that the mean length of time in repayment is 17.4 years with a standard deviation of 6.3 years. Conduct a hypothesis test to determine if the population mean time in repayment could likely be 15 years. Is this a test of one mean or proportion? State the null and alternative hypotheses. H 0 : ____________________ H a : ____________________ Is this a right-tailed, left-tailed, or two-tailed test? What symbol represents the random variable for this test? In words, define the random variable for this test. Is the population standard deviation known and, if so, what is it? Calculate the following: x ¯ = _____________ s = ____________ n = ____________ Which test should be used? State the distribution to use for the hypothesis test. Find the p -value. At a pre-conceived α = 0.05, what is your: Decision: Reason for the decision: Conclusion (write out in a complete sentence): Homework The National Institute of Mental Health published an article stating that in any one-year period, approximately 9.5% of American adults suffer from depression or a depressive illness. Suppose that in a survey of 100 people in a certain town, seven of them suffered from depression or a depressive illness. Conduct a hypothesis test to determine if the true proportion of people in that town suffering from depression or a depressive illness is lower than the percent in the general adult American population. Is this a test of one mean or proportion? State the null and alternative hypotheses. H 0 : ____________________ H a : ____________________ Is this a right-tailed, left-tailed, or two-tailed test? What symbol represents the random variable for this test? In words, define the random variable for this test. Calculate the following: x = ________________ n = ________________ p ′ = _____________ Calculate σ x = __________. Show the formula set-up. State the distribution to use for the hypothesis test. Find the p -value. At a pre-conceived α = 0.05, what is your: Decision: Reason for the decision: Conclusion (write out in a complete sentence): Level of Significance of the Test probability of a Type I error (reject the null hypothesis when it is true). Notation: α. In hypothesis testing, the Level of Significance is called the preconceived α or the preset α. p -value the probability that an event will happen purely by chance assuming the null hypothesis is true. The smaller the p -value, the stronger the evidence is against the null hypothesis.

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