CLT 1 Class.notebook. May 13, May 9 2:19 PM. May 9 2:25 PM

Transcription

1 May 9 2:19 PM May 9 2:25 PM 1

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3 Central Limit Theorem Parameter vs Statistic Law of Large Numbers Standard Error Calculating Probabilities: Suppose a forest has a mean trunk diameter of 14 inches with standard deviation 4.5 inches. Also assume that trunk diameters are normally distributed. Calculate: Probability of a randomly selected tree having a diameter of 20 inches. Probability of ten randomly selected trees having a mean diameter of 20 inches. Oct 21 7:36 AM Central Limit Theorem Symbol Statistic Symbol Parameter Sample Mean Sample Standard Deviation Sample size Population Mean Population Standard Deviation Population size Oct 21 7:36 AM 3

4 Central Limit Theorem Law of Large Numbers The larger the sample size, the closer the sample mean comes to matching the population mean. (Closer... but never the same.) Oct 21 7:36 AM x x x Oct 23 8:31 AM 4

5 May 9 3:26 PM Calculator Skills Move to PRB Press 5 or move to 5 and press Enter Press MATH Press Enter Press Enter again and again... To change the seed? 5 Sto rand. Apr 25 9:06 AM 5

6 Central Limit Theorem Defined! The distribution of sample means, x bar, is a normal distribution with mean mu and standard error σ/ n when n is sufficiently large! How large is sufficiently large? Oct 21 7:43 AM Central Limit Theorem Demo Oct 21 7:43 AM 6

7 Central Limit Theorem Standard Error A measurement of the of randomly selected means. The size of the standard error depends on: 1) 2) Measurement of variability of population: Measurement of variability of sample: Measurement of variability of sample means of sample size n: Name Oct 21 7:36 AM Calculating Probabilities: Suppose trunk diameters (at breast height) of trees in a forest are normally distributed with mean diameter 14 inches and standard deviation 4.5 inches. Suppose 9 trees are randomly selected. Population mean: Population std. deviation: Sample size: Sample mean: Standard Error (SE): Population mean of sample means: Sampling distribution of individual trees (x) : N( ) Sampling distribution of sample means? N( ) What is the probability of getting a sample mean greater than 16 inches? Apr 25 9:25 AM 7

8 Calculating Probabilities: Suppose the trees in the forest are a mix of old growth trees and younger second growth trees. Suppose trunk diameters (at breast height) of trees in the forest have a mean diameter 16 inches and standard deviation 5.8 inches. What would the population distribution of trunk heights look like? Suppose 16 trees are randomly selected and have a mean diameter greater than 18 inches. What is the probability of getting the sample mean of 16? How can we fix the problem? Apr 25 9:25 AM Activity CLT 1. Assume a population is N(120, 40). What is the probability that a randomly selected experimental unit from this population is: a. Greater than 150? b. Between 100 and 168? Nov 1 7:21 AM 8

9 Activity CLT 2. Assume a population is N(18, 5). What is the value of x such that 75% of the data is above it. Oct 30 10:20 AM Activity CLT 3. Assume that the weight of a bag of carrots is supposed to have a mean of 16 oz. But we know that the bags vary, standard deviation = What is the probability of a randomly selected bag of carrots being less than 14.3 oz. What assumption do we need to make? Oct 30 10:20 AM 9

10 Activity CLT 4. My twin brother, Rob 2, collected 100 samples from a population and then calculated the 100 sample means. He claims that the mean of these sample means would be the same as the population mean. Is he right? If yes, explain. If not, what did he really want to say? 5. The variation in the distribution of all possible sample means depends on two values. What are they? Oct 30 10:20 AM Activity CLT 6. Assume that the weight of a bag of carrots is supposed to have a mean of 16 oz. But we know that the bags vary, standard deviation = A total of 30 randomly selected bags are to be weighed. What is the probability that the sample mean weight will be: a. Less than 15 oz? b. Between 15 and 17 oz. Oct 30 10:26 AM 10

11 Activity CLT 7. During its manufacturing process, Coke fills its 12 oz bottles using an automated filling machine. This machine is not perfect and will not always fill each bottle with exactly 12 fl oz of soft drink. The amount of soft drink poured into each bottle follows a normal distribution with mean 12 fl oz and standard deviation 0.36 fl oz. A sample of 16 bottles were randomly selected and the amount of soft drink in each bottle was measured. a. What are the values of each: Pop mean Pop Std. Dev Sample size Standard error SDSM b. What is the probability that the sample mean will be less than 11.80? c. Calculate the value of the sample mean for which 30% of samples have a sample mean less than this value. Oct 30 10:27 AM Activity CLT 8. The smaller a population mean, the smaller the variation in sample means for samples from that population. T/F? Oct 30 10:27 AM 11

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13 9. A census of all LBCC transfer students found that out of 5356 students 1277 were also taking classes at OSU. A recent survey of 300 students was taken. a. What is the population proportion, π? b. What is standard deviation of sampling distribution of sample proportions? Oct 30 11:59 AM 10. A recent newspaper article claimed that 16% of all sick days throughout the year are taken illegitimately. In response to this article, the Mean Corporation has randomly selected 110 employees for an annual review of the corporation's truancy rates. The officer carrying out the review has declared: "If the proportion of illegitimate sick days per year taken by you people is not less than 0.14, you will all be fired!" Calculate the probability that the proportion of illegitimate sick days taken within the sample is less than Oct 30 12:00 PM 13

14 11. A city has 1,097,275 residents. A recent census showed that 332,789 of these residents regularly use the city's public transportation system. A survey is being conducted in which 1,076 of the city's 1,097,275 residents will be randomly selected. This question relates to the number of people in the survey that is made up of people that do use the city's public transport. The number of people in the survey that do use public transport approximately follows a normal distribution. Calculate the standard deviation of this distribution. Oct 30 12:01 PM 12. Todd Boaden is a professional skateboarder who demands that his wheels are exactly inches wide. Unfortunately the manufacturing of skateboarding wheels is flawed and wheels that are labeled inches may not necessarily be inches wide. If Todd receives a batch with more than 16% of the wheels being incorrectly labeled he becomes very angry. The proportion of wheels that are labeled inches but are not actually inches wide is equal to Calculate the probability (a) that Todd becomes very angry with the next batch of 55 wheels that he receives. You may find this standard normal table useful. Give your answer as a decimal to 4 decimal places. Oct 30 12:17 PM 14

15 Inference What is our goal? We know the population mean and standard deviation. Based on a sample we want to estimate the population mean with a certain amount of confidence. Determine N( ) for the sampling distribution of sample means. Question: What is the probability of a sample mean greater than? Question: What is a 95% confidence interval estimate of the population mean. Based on a sample we want to test an assumption of the population mean. Question: Do you reject or fail to reject the assumed mean. Oct 30 10:34 AM Oct 30 11:26 AM 15

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