8.3 Estimating a Population Mean Objectives SWBAT: STATE and CHECK the Random, 10%, and Normal/Large Sample conditions for constructing a confidence interval for a population mean. EXPLAIN how the t distributions are different from the standard Normal distribution and why it is necessary to use a t distribution when calculating a confidence interval for a population mean. DETERMINE critical values for calculating a C% confidence interval for a population mean using a table or technology. CONSTRUCT and INTERPRET a confidence interval for a population mean. DETERMINE the sample size required to obtain a C% confidence interval for a population mean with a specified margin of error.

Calculator BINGO! Let’s use the Rossman/Chance applet to simulate the calculator activity. (from pg 509). Did one method do a better job of capturing the true mean? Go to Click on Rossman/Chance Applet Collection. Under “Sampling Distribution Simulations” click on “Simulating Confidence Intervals for Population Parameter” Under “Method” select Means, Normal, z with sigma Enter mu of 100, sigma of 40, n of 4, and change conf level to 99%. Calculate a few intervals, then change intervals to 1000, or even How often is our interval containing mu?

When should we use a t* critical value rather than a z* critical value for calculating a confidence interval for a population mean? How do we calculate the value of t* to use? How do we calculate degrees of freedom? We have to use our t table using df = n – 1, and looking up the area in one-tail. If we have a sample size of 12, you would look up df of 11. If we have a sample size of 48, your df would be 47, but the table does not have a value for 47, so you need to be more conservative and use df = 40. You can also use the command invT(area:, df: ). The area is the area to the left of a desired critical value. For example, if constructing a 95% confidence interval, you would enter area: (same value you’d look up if using the table).

A Carucci sidebar: What are degrees of freedom? The number of degrees of freedom for a collection of sample data is the number of sample values that can vary after certain restrictions have been imposed on all data values. For example, if 10 students have quiz scores with a mean of 80, we can freely assign values to the first 9 scores, but the 10th score is then determined. The sum of the 10 scores must be 800, so the 10 th score must equal 800 minus the sum of the first 9 scores. Because those first 9 scores can be freely selected to be any values, we say that there are 9 degrees of freedom available.

What is a t distribution anyway? Describe the shape, center, and spread of the t distributions. Note: t does not stand for Taylor. Shape: symmetric, unimodal, but not quite Normal. It has heavier tails. The t distribution approaches the Normal distribution as df increase. Center: 0, since t is a standardized score Spread: greater than a standard Normal distribution, but gets closer to Normal as the df increase. This means we need to go farther than 1.96 SD to have 95% confidence.

What are the three conditions for constructing a confidence interval for a population mean? As with proportions, you should check some important conditions before constructing a confidence interval for a population mean. Conditions For Constructing A Confidence Interval About A Mean Random: The data come from a well-designed random sample or randomized experiment. o 10%: When sampling without replacement, check that Normal/Large Sample: The population has a Normal distribution or the sample size is large (n ≥ 30). If the population distribution has unknown shape and n < 30, use a graph of the sample data to assess the Normality of the population. Do not use t procedures if the graph shows strong skewness or outliers.

What is the formula for the standard error of the sample mean? How do you interpret this value? Is this formula on the formula sheet? ^ interpretation Formula sheet? Taylor Swift must have gotten to the formula sheet before us and shaken this formula off

What is the formula for a confidence interval for a population mean? Is this formula on the formula sheet? One-Sample t Interval for a Population Mean When the conditions are met, a C% confidence interval for the unknown mean µ is where t* is the critical value for the t n-1 distribution with C% of its area between −t* and t*.

a) Construct and interpret a 95% confidence interval for the mean weight of an Oreo cookie.

Do: Because there are 36-1 = 35 degrees of freedom and we want 95% confidence, we will use the t table and a conservative degrees of freedom of 30 to get a critical value of t* = (use tail probability 0.025). On the calculator press STAT > Tests > 8: Tinterval Stay on STATS and enter a mean of , SD of , n: 36, C-Level: 0.95 You get (11.364, 11.42)

Conclude: We are 95% confident that the interval from grams to grams captures the true mean weight of an Oreo cookie. b) On the packaging, the stated serving size is 3 cookies (34 grams). Does the interval in part (a) provide convincing evidence that the average weight of an Oreo cookie is less than advertised? Explain. The stated serving size is 34 grams for 3 cookies, or 34/3 = grams/cookie. Because all of the plausible values in the interval are greater than grams, there is no evidence that the average weight of an Oreo cookie is less than advertised. In fact, there is convincing evidence that the average weight is greater than advertised!

How can you lose credit for the Normal/Large Sample condition on the AP Exam? Not including a graph of the sample data It is not enough just to make a graph of the data on your calculator when assessing Normality. You must sketch the graph on your paper to receive credit. You don’t have to draw multiple graphs – any appropriate graph will do. Not understanding that the condition is about the population What should you do if you think the Normal/Large Sample condition isn’t met? You need to check whether it’s reasonable to believe that the population distribution is Normal. You can make a boxplot or dotplot of the data to check for Normality of the population (see example on pg ). Can you use your calculator for the Do step? Are there any drawbacks to this? You may use your calculator to compute a confidence interval on the AP exam, but there is a risk involved. If you just give the calculator answer with no work you’ll get either full credit for the “Do” step (if the interval is correct) or no credit (if it’s wrong). If you opt for the calculator-only method, be sure to name the procedure (e.g., one sample Tinterval) and to give the interval (e.g., to 11.42). If you want to be able to get partial credit, show work.

On the calculator: enter the data into a list Use the command Tinterval, switch to Data, put your list, Freq: 1, C-Level:.99 (22.991, )

Conclude: We are 99% confident that the interval from squares to squares captures the true mean number of squares of generic toilet paper needed to absorb 1/4 cup of water.

How can we choose an appropriate sample size when we plan to calculate a confidence interval for a mean?