CHAPTER 23 Inference for Means

I Need My Morning Coffee!!! A coffee machine dispenses coffee into paper cups. You’re supposed to get 10 ounces of coffee, but the amount varies slightly from cup to cup. Here are the amounts measured from a random sample of 20 cups. Is there evidence that the machine is shortchanging customers? 9.9 9.7 10.0 10.1 9.9 9.6 9.8 9.8 10.0 9.5 9.7 10.1 9.9 9.6 10.2 9.8 10.0 9.9 9.5 9.9

I Need My Morning Coffee!!! Step 1: Identify population Parameter, state the null and alternative Hypotheses, determine what you are trying to do (and determine what the question is asking). We want to know whether a particular coffee machine is shortchanging its customers. The parameter of interest is the mean amount of coffee dispensed by this machine. We assume that the mean amount is 10 ounces. The mean amount of coffee dispensed is 10 oz. The mean amount of coffee dispensed is less than 10 oz.

I Need My Morning Coffee!!! Step 2: Verify the Assumptions by checking the conditions Independence Assumption Randomization condition: We are told that the sample was randomly selected 10% condition: We can reasonably assume that we observed fewer than 10% of the cups of coffee dispensed by this machine. Plausible independence condition: There is no reason to believe that independence is violated

I Need My Morning Coffee!!! Step 2: Verify the Assumptions by checking the conditions Normality Assumption Nearly normal condition: Since 20 is relatively small, we need to check the sample distribution: Depending on how you set up your window, you may get one of the following: Using Zoom Stat (9) Adjust the Xscl to 0.1 or Both look roughly unimodal and symmetric, so it’s safe to say that the sampling distribution will be approximately normal.

I Need My Morning Coffee!!! Step 3: If the conditions are met, Name the inference procedure, state the Test statistic, and Obtain the p-value: Name the test: We will perform a 1-sample t-test Test Statistics: Obtain the p-value:

I Need My Morning Coffee!!! Step 4: Make a decision (reject or fail to reject H0). State your conclusion in context of the problem using p-value: The p-value is so small, 0.0012, that we reject the null hypothesis in favor of the alternative at the 0.05 alpha level. In other words, there is very strong evidence that the mean amount of coffee dispensed by this machine is less than the stated 10 ounces.

More Coffee, Please!!! Now that we know that this machine is ripping us off, estimate how much it is shortchanging its customers with 95% confidence.

More Coffee, Please!!! Construct a 95% confidence interval for μ, the mean of the population, from which the sample is drawn. Step 1: First, state what you want to know in terms of the Parameter and determine what the question is asking We wish to estimate the true mean amount, μ, of coffee that the machine is dispensing.

More Coffee, Please!!! Construct a 95% confidence interval for μ, the mean of the population, from which the sample is drawn. Step 2: Second, examine the Assumptions and check the conditions. Independence: Randomization condition: The cups of coffee were randomly selected 10% condition: We safely assume that we have less than 10% of all the coffee dispensed Plausible independence condition: There is no reason to believe that independence is violated

More Coffee, Please!!! Construct a 95% confidence interval for μ, the mean of the population, from which the sample is drawn. Step 2: Second, examine the Assumptions and check the conditions. Normality: Nearly normal condition: Since 20 is relatively small, we need to check the sample distribution: This looks roughly unimodal and symmetric, so it’s safe to say that the sampling distribution will be approximately normal.

More Coffee, Please!!! Construct a 95% confidence interval for μ, the mean of the population, from which the sample is drawn. Step 3: Third, Name the inference, do the work, and state the Interval. We will construct a 95% 1-sample t-Interval for means:

More Coffee, Please!!! Construct a 95% confidence interval for μ, the mean of the population, from which the sample is drawn. Step 4: Fourth, last but not least, state your Conclusion in context of the problem We are 95% confident that the machine dispenses an average of between 9.75 to 9.94 ounces of coffee per cup.

Example As always, you can do all of Step 3 in your calculator. Although the calculator will do Step 3, you still need to Steps 1, 2, and 4 on your own!!! What if we have no data? We can compute a CI or HT using the sample’s mean and standard deviation. In other words, we can use Stats rather than Data in the Inference function of the calculator. Let’s look at another example.

Fishing For a Good Fishing Line Suppose you take an SRS of 53 lengths of an 85 lb. fishing line. Your sample has an average strength of 83 lbs. with a standard deviation of 4 lbs. Determine if the fishing line should actually be considered an 85 lb. line.

Fishing For a Good Fishing Line Step 1: Identify population Parameter, state the null and alternative Hypotheses, determine what you are trying to do (and determine what the question is asking). We want to know whether a particular finishing line should be considered an 85 lb. line. The parameter of interest is the mean weight that the fishing line can hold. We assume that the mean weight that the line can hold is 85 lbs. The mean weight held is 85 lbs. The mean weight held is not 85 lbs.

Fishing For a Good Fishing Line Step 2: Verify the Assumptions by checking the conditions Independence Assumption Randomization condition: We are told that the sample is an SRS 10% condition: We can reasonably assume that we observed fewer than 10% of all lengths of 85 lb. fishing line. Normality Assumption Nearly normal condition: Since a sample of 53 is relatively large, we can say that the sampling distribution will be approximately normal by the CLT.

Fishing For a Good Fishing Line Step 3: If the conditions are met, Name the inference procedure, state the Test statistic, and Obtain the p-value: Name the test: We will perform a 1-sample t-test Test Statistics: Obtain the p-value:

Fishing For a Good Fishing Line Step 4: Make a decision (reject or fail to reject H0). State your conclusion in context of the problem using p-value: The p-value is so small, .000628, that we would rarely see such values from sampling error, so we reject the null hypothesis in favor of the alternative at the 0.05 alpha level. In other words, there is very strong evidence that the mean weight that the fishing line can use is not 85 lbs; the fishing line should not be considered an 85 lb. line.

Fishing For a Good Fishing Line Suppose you take an SRS of 53 lengths of an 85 lb. fishing line. Your sample has an average strength of 83 lbs. with a standard deviation of 4 lbs. Now make a 95% confidence interval for the mean strength of this type of fishing line.

Fishing For a Good Fishing Line Step 1: First, state what you want to know in terms of the Parameter and determine what the question is asking We wish to estimate the true mean strength of a certain type of fishing line with 95% confidence; we will produce a 95% confidence interval. Step 2: Second, examine the Assumptions and check the conditions. These are shown to be satisfied in the previous problem.

Fishing For a Good Fishing Line Step 3: Third, Name the inference, do the work, and state the Interval. We will use a 1-sample t-interval for the mean We will use the t-distribution with (n – 1) = 52 degrees of freedom

Fishing For a Good Fishing Line Step 4: Fourth, last but not least, state your Conclusion in context of the problem We are 95% confident that the true mean strength of the fishing line is between 81.9 and 84.1 pounds.

Assignment Chapter 23 Lesson: Inference for Means Read: Problems: 1 - 31 (odd)