9.1 Basic Principles of Hypothesis Testing


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1 9. Basic Principles of Hypothesis Testing Basic Idea Through an Example: On the very first day of class I gave the example of tossing a coin times, and what you might conclude about the fairness of the coin depending on the outcome of this experiment. If you got 55 heads, would you conclude that the coin was not fair? What if you got 95 heads? What if you got 75 heads? The further away our sample is from the expected value, the more suspicious we would be that the coin is not fair. Now having some knowledge about probabilities, you could find out how unlikely it is to get 75 heads or more out of if in fact the coin was fair (the probability is zero). So in a case like that, we could conclude that the coin must not be fair. In chapter 9 we will use hypothesis testing to make decisions about a population parameter ( µ or p ) based on the value of a sample statistics. The idea behind hypothesis testing is the same as the idea behind a criminal trial. The person is presumed to be not guilty until there is sufficient evidence to declare the person guilty. A null hypothesis, denoted H, states that some parameter is equal to a specific value, ex. H : µ = 27 (always use an equal sign for H ). The null hypothesis is a claim about a population parameter that is assumed to be true until declared false. An alternative hypothesis, denoted H, states that the value of the parameter differs from that specified by the null hypothesis, ex. H : µ 27 (a twotailed test), or H : 27 µ > (a righttailed test), or H : 27 µ < (a lefttailed test). The alternative hypothesis is a claim about a population parameter that will be true if the null hypothesis is false. Examples: Null hypothesis: H : The coin is fair (.5 Alternative hypothesis: H : The coin is not fair ( p.5 or p.5 or p.5 Null hypothesis: H : The person is not guilty Alternative hypothesis: H : The person is guilty Consider the M&M experiment that we did in class. What were the hypotheses? Null hypothesis: Alternative hypothesis:
2 Conclusions We reject H if the value of our test statistics falls too far away from the assumed population parameter. This is the same as saying that we support the H claim. We usually write this conclusion as "There is sufficient evidence to show that... H... ", where H is stated in words. We fail to reject H, if the difference between the sample statistics and the assumed population parameter is small, since the difference may be due to chance. We usually write this conclusion as "There is not sufficient evidence to show that... H... ", where H is stated in words. Note how we don't say that H is true. But we could say that " H may be true". Errors Our conclusion will be to either reject the null hypothesis or fail to reject it. Such conclusions are sometimes correct and sometimes not. There are two types of errors: Type I error occurs when rejecting the null hypothesis when it is actually true. The probability of making a type I error is α, the significance level. ex. Person is found guilty in court even though he did not commit the crime ex. Coin was found to be unfair even though it is fair. Type II error occurs when failing to reject the null hypothesis, when it is actually false. The probability of making a type II error is β (beta). ex. Person is found not guilty in court even though he did commit the crime. ex. Coin was found to be fair even though it is unfair. Fill each cell in the table with one of the following: Type I Error, Type II Error, or Correct Decision. Decision Reject H The Truth H is true H is false Do not reject H
3 . Test if the mean weight of women who won Miss America titles is still equal to 2 lb, or if it has changed. State hypotheses: Is it a lefttailed, righttailed, or twotailed test? If the null hypothesis is rejected, state an appropriate conclusion: What kind of error could we have made? 2. Plain M&M candies supposedly haves a mean weight that is.8535g. Test if the candies' weight is less than that. State hypotheses: Is it a lefttailed, righttailed, or twotailed test? If the null hypothesis is not rejected, state an appropriate conclusion: What kind of error could we have made? 3. Test if more than 25% of Internet users pay bills online. State hypotheses: Is it a lefttailed, righttailed, or twotailed test? If the null hypothesis is not rejected, state an appropriate conclusion: What kind of error could we have made? 4. Test if the mean hours spent per week on house chores by all housewives is less than 3. State hypotheses: Is it a lefttailed, righttailed, or twotailed test? If the null hypothesis is rejected, state an appropriate conclusion: What kind of error could we have made?
4 9.2  Hypothesis Tests About µ when σ is Known The Test Statistic is a value used in making a decision about the null hypothesis, and it is found by converting 2 the sample statistic ( x, ˆp, or s) to a score (such as z, t, or χ ) with the assumption that the null hypothesis is true. The probability that we use to determine whether an event is unusual is called the significance level of the test, and is denoted by α. If we reject H after choosing a significance level α, we say that the result is statistically significant at the α level (note that statistically significant doesn't always mean practically significant). We also say that H is rejected at the α level. The Pvalue is the probability of getting a value of the test statistic that is at least as extreme as the one representing the sample data, assuming that the null hypothesis is true. "At least as extreme" means " as far away from the expected parameter as we got, or even further". If the Pvalue is very small, it means that the probability of getting what we got in our sample is very small, which may indicate that our original assumption H is not true. H is rejected if the Pvalue is smaller than the significance level α. In a left tailed test: Pvalue = area to the left of the test statistic Pvalue Test Stat In a right tailed test: Pvalue = area to the right of the test statistic Pvalue In a two tailed test: Pvalue = twice the area in the tail beyond the test statistic Pvalue / 2 Test Stat Pvalue is the total area of both tails Pvalue / 2 Test Stat or Test Stat
5 Hypothesis Tests About µ when σ is Known: Requirements. The value of the population standard deviation σ is known. 2. Either or both of these conditions is satisfied: The population is normally distributed or n > 3. If above is satisfied we know that the sampling distribution of x is Normal. Test Statistics: x µ z = where σ x = σ x σ n 8Step Procedure for Performing a Hypothesis Test: ) State the null and alternative hypotheses of the test. 2) Choose and/or state the significance levelα. 3) State type of test, the standardized sampling distribution that should be used, and check that all of the required assumptions for using that distribution are satisfied. 4) Compute the test statistic. 5) Draw a picture of the standardized sampling distribution you are using. Label the test statistic. 6) Calculate the Pvalue. 7) Interpret the Pvalue and make a decision. If Pvalue < α then we reject the null hypothesis, and we have sufficient evidence for the alternative hypothesis. If Pvalue > α then we do NOT reject the null hypothesis, and we do NOT have sufficient evidence for the alternative hypothesis. 8) State a conclusion in the form of a detailed sentence that addresses the alternative hypothesis. When we Reject H, we say there is sufficient evidence to show that H, where H is stated in words. When we Fail to Reject H, we say there is not sufficient evidence to show that H, where H is stated in words.
6 When presenting the results of a hypothesis test, one should report the Pvalue or the value of the test statistics. That way the reader can tell exactly how likely or unlikely the test statistics was, and he/she can determine whether H could be rejected at a different level. Relationship between hypothesis tests and confidence intervals: If we test H : µ = µ vs. H: µ µ, then if the (α )% confidence interval contains µ, then H will not be rejected at the α level if the (α )% confidence interval does not contain µ, then H will be rejected at the α level P(type I error) = α and P(type II error) = β The smaller the probability of a type I error becomes, the larger the probability of a type II error becomes. Examples:. (a) The print on the package of watt General Electric softwhite lightbulbs says that these bulbs have an average life of 75 hours. Assume that the lives of all such bulbs have a normal distribution with a standard deviation of 55 hours. The mean life of a simple random sample of 25 such bulbs was 725 hours. We are concerned that the stated average life on the package is exaggerated. Test at 5% significance level if this is true. (b) Would your conclusion be different if the significance level was %?
7 2. A certain type of children's pain reliever states that it contains 325 mg of acetaminophen in each ounce of the drug. If 7 one ounce samples are tested for acetaminophen and it is determined that the sample mean is 39 mg of the drug and a population standard deviation of 26 mg. With a =., test the claim that the population mean is equal to 325 mg. We can check our answers by using programs in our calculators. Press STAT and go to TESTS
8 3. A random sample of 6 secondgraders in a certain school district are given a standardized mathematics skills test. The sample mean score is 52. Assume the standard deviation of test scores is 5. The nationwide average score on this test is 5.The school superintendent wants to know whether the secondgraders in her school district have greater math skills than the nationwide average. Use a. level of significance to test this. We never accept the null hypothesis Note, that if we Fail to Reject H, we never say we accept H because the sample data is evidence against H (in favor of H ). It is not evidence in favor of H. We cannot prove something is true when we had to assume it was true to get the test started. ex. In the famous O.J. Simpson trial there was not sufficient evidence to show that O.J. Simpson was guilty of murder. That does not mean he was innocent.
9 4. When 4 people used the Atkins diet for one year, their mean weight change was 2. lb (new weight  old weight). Assume that the standard deviation of all such weight changes is σ =4.8 lb and use a.5 significance level to test the claim that the mean weight change is less than. Based on the results, does the diet appear to be effective? Does the mean weight change appear to be substantial enough to justify the special diet?
10 9.3  Hypothesis Tests About µ when σ is Not Known Requirements. The value of the population standard deviation σ is NOT known. 2. Either or both of these conditions are satisfied: The population is normally distributed or n > 3. If above is satisfied we know that the sampling distribution of x is Normal. x µ Test Statistics: t = where sx s x = s n The 8step procedure for performing a hypothesis test:. State the null and alternative hypotheses. 2. State significance level α 3. State which test to use and why. 4. Calculate the test statistic. 5. Draw a picture and label the test statistic. 6. Calculate the pvalue. 7. Interpret pvalue and make a decision. 8. State your conclusion.. A softdrink manufacturer claims that its 2ounce cans do not contain, on average, more than 3 calories. A random sample of 64 cans of this soft drink, which were checked for calories, contained a mean of 32 calories with a standard deviation of 3 calories. Does the sample information support the alternative hypothesis that the manufacturer's claim is false? Use a significance level of 5%.
11 2. The insurance Institute for Highway Safety conducted tests with crashes of new cars traveling at 6mi/h. Total cost of the damage was found. Results are listed below for a simple random sample of the tested cars. Assume the population has a distribution that is approximately normal. $7448 $49 $95 $6374 $4277 Use a. significance level to test the claim that when tested under the same conditions, the damage costs for the population of cars have a mean of $5. 3. A simple random sample of 4 recorded speeds (in mph) is obtained from cars traveling on a section of Highway 45 in Los Angeles. The sample has a mean of 68.4 mph and a standard deviation of 5.7 mph. Use a.5 significance level to test the claim that the mean speed of all cars is greater than the posted speed limit of 65 mph.
12 9.4  Hypothesis Tests About a Population Proportion Requirements. We have a simple random sample 2. The population is at least 2 times as large as the sample. 3. The items in the population are divided into two categories. 4. The sample must contain at least individuals in each category. If above is satisfied we know that the sampling distribution of ˆp is approximately Normal. pˆ p p ( p) Test Statistics: z = where σ pˆ = σ n pˆ The 8step procedure for performing a hypothesis test:. State the null and alternative hypotheses. 2. State significance level α 3. State which test to use and why. 4. Calculate the test statistic. 5. Draw a picture and label the test statistic. 6. Calculate the pvalue. 7. Interpret pvalue and make a decision. 8. State your conclusion.. Trials in an experiment with a polygraph (lie detector) has 98 results that include 24 cases of wrong results and 74 cases of correct results. Use a.5 significance level to test the claim that such polygraph results are correct less than 8% of the time. Based on the results, should polygraph test results be prohibited as evidence in trials?
13 2. (a) 9yearold Emily Rosa chose the topic of touch therapy for a science fair project. She convinced 2 experienced touch therapists to participate in a simple test of their ability to detect human energy field. Emily constructed a cardboard partition with two holes for hands. Each touch therapist would put both hands through the two holes, and Emily would place her hand just above one of the therapist's hands and ask the therapist to identify the hand that Emily had selected. Emily used a coin toss to randomly select which hand to use. This test was repeated 28 times. If the touch therapists really did have the ability to sense a human energy field, they should have identified the correct hand much more than 5% of the time. If they just guessed, they should have been correct about 5% of the time. The touch therapists identified the correct hand 23 times out of the total 28 times, which is a success rate of 44%. Use a.5 significance level to test the claim that touch therapists are correct less than 5% of the time. (b) Would a different significance level change your conclusion in part (a)? (c) Can you argue against the validity of this study?
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