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Exploring The Central Limit Theorem and Sample Proportions in Fathom Find the true proportion of 4 year old pennies from the population of pennies p = .053 This value is called a parameter since is describes the population Use samples of size n = 10 to plot part of the sampling distribution for samples of size n = 10. In each sample, calculate the proportion of 4 year old pennies. Use 1000 samples of size n = 10. 1.) Paste your graph below. Also paste the summary table below. 2.) Theoretical mean and standard deviation for sampling distributions of sample proportions: p = the proportion of 4 year old pennies in a random sample of 10 pennies µ p = p = .053 σ p = p(1 − p) .053(1 − .053) = = .071 n 10 3.) What is the probability that a random sample of 10 pennies will contain less than 3% of 4 year old pennies in the sample? P( p < .03 ) SINCE THE SAMPLING DISTRIBUTION FOR p IS SKEWED RIGHT, WE CANNOT USE NORMAL CURVE CALCULATIONS TO FIND THIS PROBABILITY Conditions For normality 1.) np ≥ 10 (10)(.053) = .53 is not greater than or equal to 10, so the first condition for normality is not met 2.) n(1-p) ≥ 10 10(1-.053) = 9.47 is not greater than or equal to 10, so the second condition for normality is not met Exploring The Central Limit Theorem and Sample Proportions in Fathom Use samples of size n = 100 to plot part of the sampling distribution for samples of size n = 100. In each sample, calculate the proportion of 4 year old pennies. Use 1000 samples of size n = 100. 1.) Paste your graph below. Also paste the summary table below. 2.) Theoretical mean and standard deviation for sampling distributions of sample proportions: p = the proportion of 4 year old pennies in a random sample of 100 pennies µ p = p = .053 p(1 − p) .053(1 − .053) = = .022 n 100 3.) What is the probability that a random sample of 100 pennies will contain less than 3% of 4 year old pennies in the sample? P( p < .03 ) SINCE THE SAMPLING DISTRIBUTION FOR p IS SKEWED RIGHT, σ p = WE CANNOT USE NORMAL CURVE CALCULATIONS TO FIND THIS PROBABILITY Conditions For normality 1.) np ≥ 10 (100)(.053) = 5.3 is not greater than or equal to 10, so the first condition for normality is not met 2.) n(1-p) ≥ 10 100(1-.053) = 9.46 is greater than or equal to 10, so the second condition for normality is not met SINCE BOTH CONDITIONS ARE NOT MET, WE CANNOT ASSUME NORMALITY FOR THE p DISTRIBUTION Exploring The Central Limit Theorem and Sample Proportions in Fathom Use samples of size n = 200 to plot part of the sampling distribution for samples of size n = 200. In each sample, calculate the proportion of 4 year old pennies. Use 1000 samples of size n = 200. 1.) Paste your graph below. Also paste the summary table below. 2.) Theoretical mean and standard deviation for sampling distributions of sample proportions: p = the proportion of 4 year old pennies in a random sample of 200 pennies µ p = p = .053 p(1 − p) .053(1 − .053) = = .016 n 200 3.) What is the probability that a random sample of 200 pennies will contain less than 3% of 4 year old pennies in the sample? P( p < .03 ) Conditions for normality : np ≥ 10 200(.053)=10.6 is greater than 10 BOTH MET n(1-p) ≥ 10 200(1.-.053)=189.4 is greater than 10 σ p = ⎛ (.053)(1 − .053) ⎞ BOTH assumptions met therefore p N ⎜ .053, ⎟ 200 ⎝ ⎠ BY THE CENTRAL LIMIT THEOREM WE CAN CALCULATE P( p < .03 ) = .0733 There is a 7.33% chance of obtaining a sample of 200 pennies in which less than 3% of the pennies are 4 years old.