Introduction to Hypothesis Testing 7.1 348 CHAPTER 7 Hypothesis Testing with One Sample What You Should Learn A practical introduction to hypothesis tests How to state a null hypothesis and an alternative hypothesis How to identify type I and type II errors and interpret the level of significance How to know whether to use a one-tailed or two-tailed statistical test and find a P@value How to make and interpret a decision based on the results of a statistical test How to write a claim for a hypothesis test HypothesisTests Stating a Hypothesis Types of Errors and Level of Significance Statistical Tests and P-Values Making a Decision and Interpreting the Decision Strategies for Hypothesis Testing Hypothesis Tests Throughout the remainder of this text, you will study an important technique in inferential statistics called hypothesis testing. A hypothesis test is a process that uses sample statistics to test a claim about the value of a population parameter. Researchers in fields such as medicine, psychology, and business rely on hypothesis testing to make informed decisions about new medicines, treatments, and marketing strategies. For instance, consider a manufacturer that advertises its new hybrid car has a mean gas mileage of 50 miles per gallon. If you suspect that the mean mileage is not 50 miles per gallon, how could you show that the advertisement is false? Obviously, you cannot test all the vehicles, but you can still make a reasonable decision about the mean gas mileage by taking a random sample from the population of vehicles and measuring the mileage of each. If the sample mean differs enough from the advertisement’s mean, you can decide that the advertisement is wrong. For instance, to test that the mean gas mileage of all hybrid vehicles of this type is m = 50 miles per gallon, you take a random sample of n = 30 vehicles and measure the mileage of each. You obtain a sample mean of x = 47 miles per gallon with a sample standard deviation of s = 5.5 miles per gallon. Does this indicate that the manufacturer’s advertisement is false? To decide, you do something unusual—you assume the advertisement is correct. That is, you assume that m = 50. Then, you examine the sampling distribution of sample means (with n = 30) taken from a population in which m = 50 and s = 5.5. From the Central Limit Theorem, you know this sampling distribution is normal with a mean of 50 and standard error of 5.52 30 ≈ 1. In the figure below, notice that the sample mean of x = 47 miles per gallon is highly unlikely—it is about 3 standard errors 1z ≈ -2.992 from the claimed mean. Using the techniques you studied in Chapter 5, you can determine that if the advertisement is true, then the probability of obtaining a sample mean of 47 or less is about 0.001. This is an unusual event. Your assumption that the company’s advertisement is correct has led you to an improbable result. So, either you had a very unusual sample, or the advertisement is probably false. The logical conclusion is that the advertisement is probably false. Hypothesized mean 46 47 48 49 50 51 52 53 54 −4 −3 −2 −1 0 1 2 3 4 x z μ = 50 Sample mean x = 47 z≈−2.99 Sampling Distribution of x Study Tip As you study this chapter, do not get confused regarding concepts of certainty and importance. For instance, even if you are very certain that the mean gas mileage of a type of hybrid vehicle is not 50 miles per gallon, the actual mean mileage might be very close to this value and the difference might not be important.
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