9.2 - Hypothesis Tests About µ when σ is Known

Transcription

1 9.2 - Hypothesis s About µ when σ is Known The istic is a value used in making a decision about the null hypothesis, and it is found by 2 converting 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 P-value 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 P-value 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 P-value is smaller than the significance level α. In a left tailed test: P-value = area to the left of the test statistic P-value In a right tailed test: P-value = area to the right of the test statistic P-value In a two tailed test: P-value = twice the area in the tail beyond the test statistic P-value / 2 P-value is the total area of both tails P-value / 2 or

2 Hypothesis s About µ when σ is Known: Requirements 1. 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. istics: x µ z = where σ x = σ x σ n 8-Step Procedure for Performing a Hypothesis : 1) e the null and alternative hypotheses of the test. 2) Choose and/or state the significance levelα. 3) e 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 P-value. 7) Interpret the P-value and make a decision. If P-value < α then we reject the null hypothesis, and we have sufficient evidence for the alternative hypothesis. If P-value > α then we do NOT reject the null hypothesis, and we do NOT have sufficient evidence for the alternative hypothesis. 8) e 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 1, where H1 is stated in words. When we Fail to Reject H, we say there is not sufficient evidence to show that H 1, where H1 is stated in words.

3 When presenting the results of a hypothesis test, one should report the P-value 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. H1 : µ µ, then if the (1-α )% confidence interval contains µ, then H will not be rejected at the α level if the (1-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: 1. (a) The print on the package of 1-watt General Electric soft-white light-bulbs 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. at 5% significance level if this is true. (b) Would your conclusion be different if the significance level was 1%?

4 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 319 mg of the drug and a population standard deviation of 26 mg. With a =.1, 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

5 3. A random sample of 6 second-graders 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 15. The nationwide average score on this test is 5.The school superintendent wants to know whether the second-graders in her school district have greater math skills than the nationwide average. Use a.1 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 1 ). 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.

6 4. When 4 people used the Atkins diet for one year, their mean weight change was -2.1 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?