Example: Awe…Nuts! - Village Christian School

How many movies do you watch?

Does the CLT apply to means?  The Central Limit Theorem states that the more samples you take, the more Normal your graph will appear for either the sample mean or sample proportion.  What do we need to verify in order to say the distribution should be approximately normal?  Independence Assumption: • In order to be able to consider each sample an unbiased estimator, we must insure that random selection was used. • If we want to insure we aren’t affecting the probability of selecting each sample, we must insure that the sample we take is less then 10% of our population.

Does the CLT apply to means?

Back to the show…

Back to the show…  Suppose that the number of movies viewed in the last year by all high school students in CA has an average of 19.3 and a standard deviation of 15.8 . If we were to randomly sample 100 high school students many times, what would we expect the average of the sampling distribution to be? What about the standard deviation?  Independence Assumption: We are given that the selection would be random. We can also safely assume that there are more then 1,000 high school students in CA.  Large Enough Condition: Since n = 100 > 30 we can state that the distribution would be approximately Normal.

Example: Awe…Nuts! At the P. Nutty Peanut Co., dry-roasted, shelled peanuts are placed in jars by a machine. The distribution of the weights in the jars is approximately Normal with a mean of 16.1 oz and a standard deviation of .15 oz.  Without doing any calculations, explain which outcome is more likely: randomly selecting a single jar and finding that it weighs less than 16 oz. or randomly selecting 10 jars and finding that the average is less than 16 oz.?  Find the probability of each of these events.

Example: Awe…Nuts! At the P. Nutty Peanut Co., dry-roasted, shelled peanuts are placed in jars by a machine. The distribution of the weights in the jars is approximately Normal with a mean of 16.1 oz and a standard deviation of .15 oz.  Without doing any calculations, explain which outcome is more likely: randomly selecting a single jar and finding that it weighs less than 16 oz. or randomly selecting 10 jars and finding that the average is less than 16 oz.?  Follow the 4 Step Process!  1. State what you want to know: We want to find the probability of selecting a single jar that weighs less then 16 oz.

Example: Awe…Nuts! At the P. Nutty Peanut Co., dry-roasted, shelled peanuts are placed in jars by a machine. The distribution of the weights in the jars is approximately Normal with a mean of 16.1 oz and a standard deviation of .15 oz.  Without doing any calculations, explain which outcome is more likely: randomly selecting a single jar and finding that it weighs less than 16 oz. or randomly selecting 10 jars and finding that the average is less than 16 oz.?  Step 2: Verify Your Assumptions  Independence Assumption: We are told that the jar is randomly selected. And we can assume that on any given day the P. Nutty Company makes more then 10 jars of peanuts.

Example: Awe…Nuts! At the P. Nutty Peanut Co., dry-roasted, shelled peanuts are placed in jars by a machine. The distribution of the weights in the jars is approximately Normal with a mean of 16.1 oz and a standard deviation of .15 oz.  Without doing any calculations, explain which outcome is more likely: randomly selecting a single jar and finding that it weighs less than 16 oz. or randomly selecting 10 jars and finding that the average is less than 16 oz.?  Step 2: Verify Your Assumptions  Large Enough Condition: Since we are given that the distribution of weights is approximately Normal, even though our sample is small, it is safe to proceed.

Example: Awe…Nuts!

Example: Awe…Nuts!

This makes sense, because if the distribution is approximately Normal. Then the mean would divide the distribution in half.

Example: Awe…Nuts! At the P. Nutty Peanut Co., dry-roasted, shelled peanuts are placed in jars by a machine. The distribution of the weights in the jars is approximately Normal with a mean of 16.1 oz and a standard deviation of .15 oz.  Without doing any calculations, explain which outcome is more likely: randomly selecting a single jar and finding that it weighs less than 16 oz. or randomly selecting 10 jars and finding that the average is less than 16 oz.?  In your notes, follow the 4 step process to solve this problem. I will randomly select students to come up to the board and show their work.