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Statistics 400 - Lecture 10 Last day: 8.3 and started 8.4 Today: Sections 8.4 Hypothesis Testing Hypothesis testing is a statistical technique to test if a conjecture about a population parameter is true Has 4 Main Steps: Null and Alternate Hypotheses Test Statistic P-Value Decision based on pre-specified error rate Example Heights of one-year-old girls normally distributed with mean 30 inches and standard deviation of 1.2 inches Company claims taking 500 mg of Vitamin C makes the girls taller Is the company’s claim true? 1. Hypotheses Hypotheses are statements about a population and is expressed in terms of the population parameters Begin by making an assumption of no change (Treatment has no effect) This statement is called the null hypothesis (H0) Test will be designed to assess evidence against H0 Hypothesis we suspect is true is called the alternate hypothesis (H1 ) Assume H0 is true, collect data and see if there is evidence against H0 and in favor of H1 Example Heights of one-year-old girls normally distributed with mean 30 inches and standard deviation of 1.2 inches Company claims taking 500 mg of Vitamin C makes the girls taller H0: H1: 2. Test Statistic Test statistic measures compatibility between H0 and the data It is based on 2 principles: based on estimate of the parameter that appears in the hypotheses measures distance of estimate from the hypothesized value When H0 is true, we expect the value of estimate to be close to parameter on average Example (continued) Suppose a random sample of 100 baby girls are given 500 mg of vitamin C daily for 1 year Mean height of the girls after 1 year is 32 inches (estimates population mean) What is the distribution of x if H 0 is true? What is the distribution of x if H 1 is true? 3. P-Value Assume that H0 is true The P-value is the the probability of observing a test statistic as extreme or more extreme than the value actually observed when H0 is true What does a small p-value imply? How small is small? Example (continued) If H0 is true, the distribution of the sample mean is: What does “extreme” mean in this context? P-value= 4. Decision How small must the p-value be to reject H0? Must decide which value of the test statistic give evidence in favor of H1 Would like the probability of observing such values to be small when H0 is true The significance level of the test is: Example (continued): P-value= Significance level: Decision: Hypothesis Testing is Similar to a Jury Trial H0: state of no change Not Guilty H1: condition believed to be true Guilty Collect data and compute test statistic Collect evidence Compute p-value Weigh evidence Reject or do not reject H0 based on significance level Decide if evidence is in favor of guilty beyond a reasonable doubt How do we interpret significance level Some common significance levels: Have we proven that H0 is true or false? Z-Test for the Population Mean Have a random sample of size n ; x1, x2, …, xn H 0 : 0 Test Statistic: Z X S/ n Can be used for normal population or for large samples (why?) Z-Test for the Population Mean (cont.) P-value depends on the alternative hypothesis: H1 : 0 : p - value P(Z z) H1 : 0 : p - value P(Z z) H1 : 0 : p - value 2P(Z | z |) Example: Scientists believe that abused children show elevated levels of depression To test this assertion, as random sample of 50 abused children were given a Profile of Moods States (POMS) test The results showed a mean depression score of 17.3 and standard deviation of 5.4 Test, at the 5% level, whether abused children have a higher mean depression that that of the general population (mean=15) Example: A study titled “St. John’s Wort: Effect on CYP3A4 Activity” (Clinical Pharmacology and Therapeutics, 2000) reported a study that assesed urinary 6-beta-horoxycortisol/cortisol ratio in 12 subjects after 14 days of therapy with St. John’s Wort. The baseline mean ratio for the target population is 7.0 and the scientists wished to determine if the therapy resulted in increased a urinary 6-beta-horoxycortisol/cortisol ratio Using the data below, test this hypothesis Patient urinary 6-betahoroxycortisol/cortisol ratio Patient urinary 6-betahoroxycortisol/cortisol ratio 1 2 3 4 5 6 16.8 13.7 11.3 20.3 7.0 6.1 7 8 9 10 11 12 5.4 14.9 9.2 6.4 12.9 7.2

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