Section 9.2 A Significance Test for a Mean DISCUSSION Statistics In section 8.2 we looked at a significance test for a proportion. We computed a test statistic (z value) to determine if a hypothesized population proportion _____ is compatible with a sample proportion ______. The test statistic value was: In this section we are going to compute a test statistic to determine if a hypothesized mean value for a population, _______, is compatible with a sample mean, ______. It would be nice if we could just use a z-distribution like we did in Section 7.2 (Sampling Distribution of a Sample Mean), where : However, in this case we do not know ______, the population standard deviation, so we must use the standard deviation of our sample, _____. Remember, though, that if we use ____ the ______ values must be adjusted, and we compute a ______ value instead, and use a t-table when working with probabilities. Recall that the probability values associated with a t-table refer to tail probabilities, or _____ values. Since there are so many parallels between sections 8.2 and 9.2, this discussion will just highlight the characteristics for a Significance Test for a Mean. You are encouraged to read through the section itself for more details. Here we will just mention the characteristics of using a P-value in a test of a mean, the components of a Significance Test for a Mean, and we will look at an example problem. Using a P-value in a Test of a Mean The ____________ is the probability of getting a random sample, from a distribution with the ________ given in the _______ hypothesis, ______, that has a value of ______ that is as _____________ or even more extreme than the _____________ value of ______ computed from the _____________. When the _____ value is close to ______, you have convincing evidence ____________ the null hypothesis. When the _____ value is large, the result from the ___________ is consistent with the hypothesized mean and you do not have enough evidence to ____________ the null. The test of ___________ for the ________ is called a ____________. Components of a Significance Test for a Mean 1.) Name the test and check conditions. Three conditions must be tested for: _____________________. For a ______________, the sample must have been randomly selected. For an ________________, the ________________ must have been ________________ assigned to the _________________ units. (1. Name the test and check conditions, continued) ____________________. The sample must look like it is reasonable to assume it came from a __________________ distributed population OR the sample ___________ must be ____________ enough for the sampling distribution of the sample mean to be approximately normal (there is no set rule for approximately normal) 2.) State your hypotheses. The _______ hypothesis is that the population mean, ______, has a particular value ______. This is typically abbreviated as ________ = _______. The _____________ hypothesis, _______, has one of three forms: 3.) Compute the test statistic, find the ____value, and draw a sketch. The ________ statistic is the distance from the sample mean, _____, to the hypothesized value, ______, measured in standard errors: (the P-value is the probability of getting a value of t that is as extreme or more extreme than the one computed form the actual sample if the null hypothesis is true). 4.) Write your _______________ linked to your computations and in the context of the problem. Reject the _______ hypothesis, ______, if your _____value is less than the level of significance, ______. If the _____value is greater than or equal to _______, do not reject the _______ hypothesis, _____. If you are not given a value of ______, assume that _____ = ______. Example (from pages 590 – 591 of text) McDonald’s “target value” for the mass of a large order of French Fries is 171 g. 30 bags of French Fries at a McDonald’s Restaurant were purchased. When plotted, the mass distribution has a reasonably normal distribution shape. The mean of this sample was ______ = 144.07 grams with a sample standard error of _____ = 12.28 grams. Based on your sample, you want to determine if McDonald’s “target value” is reasonable. Step 1.) The problem states that the shape of the sample looks approximately normal, so we assume we are OK to do the analysis. Step 2.) State the null hypothesis and alternative hypothesis. Step 3.) Compute the test statistic, find the P-value, and draw a sketch. Step 4.) Write your conclusion lined to the computations and in context of the problem.