9 8.2 Significance TestJenny and Maya wonder if heads and tails are equally likely when a penny is spun. They spin pennies 40 times and get 17 heads. Later… Miguel and Kevin spin pennies and get 10 heads out of 40 spins for a sample proportion, p-hat of A sample proportion is said to be statistically significant if it isn’t a reasonably likely outcome when the proposed standard is true.
10 P-valuesInstead of simply reporting whether a result is statistically significant, it is common practice also to report a P-value. The P-value for a test is the probability of seeing a result from a random sample that is as extreme as or more extreme than the result you got from your random sample if the null hypothesis is true. Find the P-value for Jenny and Maya. They spin pennies 40 times and get 17 heads. A large P-value tells you that the sample proportion you observed is near p0, so your result isn’t statistically significant.
11 Null and Alternative Hypothesis The null hypothesis, H0, specifies a population model parameter of interest and proposed a value for the parameter. Written as H0: parameter = hypothesized value Spinning the coin we can hypothesize that H0: p = 0.5 The alternative hypothesis, HA, contains the values of the parameter that we consider plausible if we reject the null hypothesis. (written with ≠, <, or >) The level of significance the p-value for which the null hypothesis will be rejected. α - alpha
12 To reject or not to reject In a courtroom, a defendant is either guilty or not guilty. Notice they don’t say innocent. Same with the null hypothesis. Either reject the null hypothesis or fail to reject the null hypothesis. The rejection or lack of rejection is based on the p-value and whether or not it is statistically significant.
13 Example: Finding Critical Values Suppose you want to reject the null hypothesis when the test statistic, z, is in the outer 10% of the standard normal distribution—that is, with a level of significance equal to What should you use as critical values? α = 0.1
14 Pg 498 Components of Significant Test Give the name of the test and check the conditions for its use. For a significance test for a proportion, three conditions must be met: The sample is a simple random sample from a binomial population. Both np0 and n(1 − p0) are at least 10. The population size is at least 10 times the sample size. State the hypotheses, defining any symbols. When testing a proportion, the null hypothesis, H0, isH0: The percentage of successes, p, in the population from which the sample came is equal to p0.The alternative hypothesis, Ha, can be of three forms:
15 Components continued… 3. Compute the test statistic, z, and find the critical values, z*, and the P-value. Include a sketch that illustrates the situation. 4. Write a conclusion. There are two parts to stating a conclusion: Compare the value of z to the predetermined critical values, or compare the P-value to . Then say whether you reject the null hypothesis or don’t reject the null hypothesis, linking your reason to the P-value or to The critical values
16 H.W. and upcoming datesRead all of 8.2, copy formulas and definitions. After 2nd day of 8.2, there will be a quiz on 8.1 and 8.2 that will include questions about symbols, formulas and vocabulary in addition to problems. Know the steps for conducting a significance test.
17 Form A – Ch. 7 Test Common Mistakes 2. If two values are selected randomly and independently from two populations, then which of these statements is not true?A. The mean of the sampling distribution of the sum of the values is equal to thesum of the population means.B. The mean of the sampling distribution of the difference of the values is equalto the difference of the population means.C. The variance of the sampling distribution of the sum is equal to the sum of thevariances.D. The variance of the sampling distribution of the difference is equal to thedifference of the variances.E. If the two populations are normally distributed, so are the distributions of thesum and the difference.Related to 1 c. on the Practice Test and rules in Ch. 6 on variance and averages
18 Form A2. A study measured the waist size of 250 men finding a mean of inches and a standard deviation of 4.02 inches. Here is the histogram of the measurements.
19 Form B2. An investigator anticipates that the proportion ofred blossoms in his hybrid plants is In arandom sample of 50 of his plants, 22% of theblossoms are red. If the investigator anticipates thecorrect proportion of red blossoms, the standarderror of the sampling distribution of the sampleproportion is approximatelyA B C D E. cannot be determined
21 Extra day for Review Friday and Test on Tuesday March 20th Choice for StatisticsExtra day for Review Friday and Test on Tuesday March 20thQuiz on Friday March 16thTuesday Review, go over quizTest on Thursday March 22nd