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Name: ___________________________ Statistics Teacher: _______________ Block: ______ Central Limit Theorem Definitions Parameter: number that describes the ___________________________. It is a _____________ ___________________ but we do not know its value because ___________________________ ____________________________________________________________________________. Statistic: number the describes a ________________________. It is known when we have _____________________________________________________________ but it can change ___________________________________________________________________________. REMEMBER S tatistics come from S amples P arameters come from P opulation NOTE − Mean of a Population: ______ − Mean of a Sample: ______ More Definitions Sample means: ________________________________________________________________ _____________________________________________________________________________. Sampling Distribution: _________________________________________________________ _____________________________________________________________________________. − The variability of a statistic is described by ________________________________________ _____________________________________________________________________________. − If you increase your ___________________________________________________________ the variability in the distribution of ________________________________________________. Name: ___________________________ Statistics Teacher: _______________ Block: ______ − We are about to examine the distribution of sample means. − This is also referred to as the ____________________________________________________. Let’s experiment . . . − Bowling scores are normal distributed with a mean of 160 and standard deviation of 20. − Is it normal to bowl a 139? ______ − Is it normal to bowl a 225? ______ − Let’s have the calculator simulate some bowling scores − CALCULATOR COMMAND: MATH ⇒ PRB ⇒ #6 ⇒ randNorm(______,______) − We want to be able to look at the average score from 5 bowling games. − First let’s have the calculator simulate 5 games with just one command: − CALCULATOR COMMAND: MATH ⇒ PRB ⇒ #6 ⇒ randNorm(_____,_____,___) − Now we want to find the average score from bowling 5 random games. − CALCULATOR COMMAND: 2ND ⇒ STAT ⇒ MATH ⇒ #3 mean( − MATH ⇒ PRB ⇒ #6 mean(randNorm(_______,_______,_____)) − Round your average score of 5 games to the nearest whole number. _____________ Looking at all of our sample means in L1, what do you notice? − All of the scores were not exactly _______, however the numbers ______________________ ___________________________________ when we were just calculating random scores of 1 bowling game. ● The mean of the sample means should be equal to the ____________________________. ● The mean of the population and mean of our sample mean is ______________________. ● If we used even bigger samples with more people the mean would be _______________. What about the standard deviation? − __________________________________________________________________________. Name: ___________________________ Statistics Teacher: _______________ Block: ______ Reflect . . . In our bowling example: − What was the sample size for each sample taken? ________________ − What were the sample means? __________________________________________________ − Explain how the variability of the sample distribution was less than the variability of the population. ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ The Central Limit Theorem Given: 1. A random variable _____ has a distribution with a mean _____ and a standard deviation ____. 2. Samples of size _____ are selected from this population and the ________________________ are calculated. Conclusions The Central Limit Theorem states that if we consider all ________________________________ ________________ from a population with mean _____ and standard deviation _____ and we look at the sample means from these samples: 1. The mean of the sample mean, ______, is the ______________________________________. 2. The standard deviation of the sample mean is: 3. As the sample size _____________________________, the distribution of the sample mean will _________________________________________________________________________. Name: ___________________________ Statistics Teacher: _______________ Block: ______ Example 1: Rat weights are normally distributed with μ = 8 lbs. and σ = 3 lbs. − Identify Sample Mean: − Calculate Sample Standard Deviation − Run normalcdf 1) I randomly select 1 rat. What is the probability that it weighs less than 6 lbs.? _____________ 2) I randomly select 30 rats. Find the mean and SD of the sampling distribution of 30 rats. _____________________________________________________________________________ 3) What is the probability that the 30 rats have an average weight less than 6 lbs.? ____________ Example 2: ACT scores are normally distributed with µ = 18.6 and σ = 5.9. 1) Find the probability that a randomly selected student has a score higher than 21. ___________ 2) Find the probability that 10 randomly selected students have an average score greater than 21. ___________ 3) If 20 students are randomly selected, what is the probability that their average score is greater than 21? ___________ Example 3: Women's heights are normally distributed with mean = 65 inches and standard deviation = 2.5 inches. 1) Find the probability that a randomly selected women is between 63 and 65 inches. ___________ 2) Find the probability that 10 randomly selected women have an average height between 63 and 65 inches. ___________ Name: ___________________________ Statistics Teacher: _______________ Block: ______ 3) Find the probability that 25 randomly selected women have an average height between 63 and 65 inches. ____________ What is original population isn’t normally distributed? ______________________________________________________________________________ ______________________________________________________________________________ ____________________________________________________________________________
