Section 9.3 One-sample z and t-test for a Population Mean Hypothesis Tests Fixed Levels of Significance

This is Greek to me! The Greek letter alpha, α, is called the significance level. If the p-value is at or below the significance level, we consider the data to be statistically significant. This is often 0.05, but other times you will see other levels given to you.

Steps for a Hypothesis Test Step 1: Write the null and alternative hypotheses. ALWAYS use parameter symbols in the hypotheses. Step 2: Choose the appropriate inference procedure. Check the conditions of the test. (They will match the conditions for confidence interval). Step 3: Identify the test by name OR by formula. Calculate the test statistic and find the p-value. Be sure you graph and shade here. Step 4: Link the p-value to the conclusion and write the conclusion in the context of the problem. At this time, we’ll only be studying μ. Since we are given sigma, our procedure is the one-sample z-test. The conditions are the same as for the z-interval.

One sample z-statistic (THE FORMULA) This only works when the parameter of interest is μ.

The one-sample t-statistic The t-statistic does not follow a normal distribution. It follows a t-distribution with n – 1 degrees of freedom.

Let’s try it! A score of 275 or higher on the National Assessment of Educational Progress corresponds with the ability to balance a checkbook. In a SRS of 840 young Americans, the mean score was 272. Is this sample good evidence that the average score for young Americans is below 275? Assume σ = 60.

Back to the cola sweetness Cola makers test new recipes for loss of sweetness during storage. Here are the sweetness losses (sweetness before storage minus sweetness after storage) found by 10 tasters for one new cola recipe: {2.0, 0.4, 0.7, 2.0, -0.4, 2.2, -1.3, 1.2, 1.1, 2.3} Are these data good evidence that the cola lost sweetness?

Two-sided Alternatives When the alternative hypothesis is two-sided, finding the p-value is a little different. You don’t know which way you are looking. Because you shade on BOTH sides of the normal curve, you need to DOUBLE the p-value. Example: Suppose the mean systolic blood pressure for males is 128 with a standard deviation of 15. We feel that the mean systolic BP for male executives differs from that of the entire population. We take a sample of 72 executives and find that x-bar is 126.07. Does this give us good evidence that the executives have a mean BP that differs from that of the population?

Try this one on your own! A pharmaceutical manufacturer forms tablets by compressing a granular material that contains the active ingredient and various fillers. The hardness of a example from each lot of tablets produced is measured in order to control the compression process. The target values for the hardness are μ = 11.5 and σ = 0.2. The hardness data for a sample of 20 tablets are {11.627, 11.613, 11.493, 11.602, 11.360, 11.374, 11.592, 11.458, 11.552, 11.463, 11.383, 11.715, 11.485, 11.509, 11.429, 11.477, 11.570, 11.623, 11.472, 11.531}. Is there significant evidence at the 5% level that the mean hardness for the tablets is different from the target value?

Instructions when using your calculator Go to Stats. Choose Test. Then pick the appropriate one. μ0 is your null hypothesis. You must know σ (for the population if a z test). Enter x-bar and n. Choose what alternative hypothesis you want. If you choose Calculate, you will be given your information. If you choose Draw you will see a picture with the z/t-value and p-value. If you don’t have all the information above, you must enter your data in L1 and choose Data.

Example In a discussion of SAT scores, someone comments, “Because only a minority of high school students take the test, the scores overestimate the ability of typical high school seniors. The mean SAT mathematics score is about 475 with a standard deviation of 100, but I think that if all seniors took the test, the mean score would be less than 475.” You gave the test to an SRS of 500 seniors from California. These students had a mean score of x-bar = 461. Is this good evidence against the claim that the mean for all California seniors is less than 475?

Example Here are estimates of the daily intakes of calcium (in milligrams) for 38 women between the ages of 18 and 24 years who participated in a study of women’s bone health: {808, 882, 1062, 970, 909, 802, 374, 416, 784, 997, 651, 716, 438, 1420, 948, 1050, 976, 572, 403, 626, 774, 1253, 549, 1325, 446, 465, 1269, 671, 696, 1156, 684, 1933, 748, 1203, 2433, 1255, 1100, 1425} Display the data using a histogram and then make a normal probability plot. Describe the distribution of calcium intakes for these women. Calculate the mean and the standard deviation of the sample. Is the sample of 18-24 year olds calcium intakes significantly different from that of the population, if the populations mean was 904.5?

Homework Chapter9 # 70, 72, 74, 79, 90, 99