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What is meant by the term Statistically Significant? If the results of a study/experiment are unlikely to happen by Random Chance alone when the null hypothesis (Ho) is true. Results are statistically significant whenever they provide convincing evidence that the alternative hypothesis is true. Not 100% proof, just convincing evidence. (Think about a court room. They use “beyond a reasonable doubt” or a “preponderance of evidence”. Not 100% certainty). α, (Greek lower-case alpha) is the symbol we use as a basis for our determination of statistical significance. Usually we use an α=0.05 level of statistical significance.

9.1 – Significance Tests: The Basics Is it possible we made a mistake?

Introduction to Errors 2 Ways of Making a Wrong Decision: We can reject the null hypothesis when in fact it is true. This is called a Type I Error.

Introduction to Errors 2 Ways of Making a Wrong Decision: We can accept (fail to reject) the null hypothesis when in fact it is false. This is called a Type II Error.

State a Type I and a Type II Error. Suppose you suspect a student is cheating during a very challenging test – the most difficult test of the year. When the tests are passed back, the teacher has curved the test out of the highest grade…the suspected cheater’s! Do you come forward???

You have 4 possible outcomes: The truth is that he didn’t cheat. You decide that he didn’t cheat. All is well. Right decision. What are the other outcomes? The truth is that he did cheat. You decide he did cheat. All is well. Right decision. The truth is that he didn’t cheat (Ho). You decide that he did cheat (Ha). This is a Type I Error. The truth is that he did cheat (Ha). You decide that he didn’t cheat (Ho). This is a Type II Error.

Which is Worse? If you want to reduce the possibility of a type I error, you must choose a critical value that is further into the tail – you want to be as sure as possible that a person did it. Circumstantial evidence is not enough. You really want to be as sure as possible so that you don’t have an innocent person punished.

Which is Worse? BUT, if we do that, there are certainly going to be more people who get away with the cheating because in that process of being absolutely sure of the person’s guilt, we will be letting more people go for whom we have strong suspicion, but not positive proof. That means we are increasing the possibility of a type II error.

Which is Worse? If we want to reduce the possibility of a type II error, (we don’t want cheaters getting away with it) we need to take anyone we strongly have suspicions about cheating and punish them. BUT, if we do that, there are bound to be students who get caught by the circumstantial evidence against them and possibly get punished for cheating when they didn’t do it. Hence, more type I errors.

Remember based on the following chart Truth about the population Ho is True Ha is True Reject Ho TYPE I ERROR  Good decision Accept Ho TYPE II ERROR Decision you make based on your evidence sample

State a Type I and a Type II Error. You are thinking about opening a restaurant and are searching for a good location. From research done, you know that the mean income of those living near the restaurant must be over $45,000 to support the type of upscale restaurant you wish to open. You decide to take a SRS of 50 people living near one potential location. Based on the mean income of this sample, you will decide whether to open a restaurant there.

Are these potato chips too salty? The mean salt content of a certain type of potato chip is supposed to be 2.0 mg. The salt content of these chips varies normally with standard deviation σ = .1 mg. From each batch produced, an inspector takes a sample of 50 chips and measures the salt content from each chip. The inspector rejects the entire batch if the mean salt content is significantly more than 2 mg at the 5% significance level. Suppose our sample gives a mean salt content of 2.03 mg. P(Type I Error) = α

Power The probability of a Type II Error tells us the probability of “accepting” the null hypothesis when it is actually false. The complement of this would be the probability of not accepting the null hypothesis when it is actually false. This is the power of a significance test. 1 – P(Type II Error)  1 – β

Are these potato chips too salty? The mean salt content of a certain type of potato chip is supposed to be 2.0 mg. The salt content of these chips varies normally with standard deviation σ = .1 mg. From each batch produced, an inspector takes a sample of 50 chips and measures the salt content from each chip. The inspector rejects the entire batch if the mean salt content is significantly different from 2 mg at the 5% significance level. Suppose our sample gives a mean salt content of 2.025 mg. What if our null is wrong? What if the distribution is actually centered at 2.05 mg of salt?

The Interaction Between Type I Errors, Type II Errors, and Power… Now, since we want to maximize the probability of making a correct decision, we want to maximize power! Increase α. A tougher standard of proof (lower α) gives less power and higher risk of Type II Error…so increase α to increase power. http://www.intuitor.com/statistics/T1T2Errors.html

The Interaction Between Type I Errors, Type II Errors, and Power… Now, since we want to maximize the probability of making a correct decision, we want to maximize power! Use an alternative hypothesis that is farther away from H0. Gives more power… http://www.intuitor.com/statistics/T1T2Errors.html

The Interaction Between Type I Errors, Type II Errors, and Power… Now, since we want to maximize the probability of making a correct decision, we want to maximize power! Increase the sample size. A larger sample size decreases the standard deviation and therefore shrinks the spread of both curves. This gives more power. See Type 2 Errors program!

So what exactly are you supposed to be able to do with all of this??? Interpreting over calculating!

Pizza Hut Pizza Hut, after test-marketing the Bigfoot Pizza, concluded that introduction of the Bigfoot nationwide would increase its sales by more than 14%. This conclusion was based on recording sales information for a random sample of Pizza Hut restaurants selected for the marketing trial. State the hypotheses. State a Type I Error and its implications. State a Type II Error and its implications. Name at least one way to maximize power?

Homework Pg. 547: #19 – 25 odd