CHAPTER 7: Central Limit Theorem: CLT for Averages (Means)


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1 CHAPTER 7: Cetral Limit Theorem: CLT for Averages (Meas) X = the umber obtaied whe rollig oe six sided die oce. If we roll a six sided die oce, the mea of the probability distributio is X P(X = x) Simulatio: We simulated rollig a six sided die 100 times usig our calculators. X = the sample average for a sample of 100 rolls of oe die Whe we each simulated rollig a six sided die 100 times i class ad foud the average for each studets' sample of 100 rolls, the mea of the sample averages was Compare the spreads of the probability distributio for rollig oe die oce to the probability distributio of averages from samples of rollig the die 100 times. Which has more spread? Which is more cocetrated about the mea? Suppose that we have a large populatio with with mea ad stadard deviatio. Suppose that we select radom samples of size items this populatio. Each sample take from the populatio has its ow average x. The sample average for ay specific sample may ot equal the populatio average exactly. The sample averages x follow a probability distributio of their ow. The average of the sample averages is the populatio average: x = The stadard deviatio of the sample averages equals the populatio stadard deviatio divided by the square root of the sample size x sample size The shape of the distributio of the sample averages x is ormally distributed IF the sample size is large eough OR IF the origial populatio is ormally distributed The larger the sample size, the closer the shape of the distributio of sample averages becomes to the ormal distributio. x ~ N, This is called the Amazigly, this meas that eve if we do't kow the distributio of idividuals i the origial populatio, as the sample size grows large we ca assume that the sample average follows a ormal distributio. We ca fid probabilities for sample averages usig the ormal distributio, eve if the origial populatio is ot ormally distributed. eve if we do't kow the shape of the distributio of the origial populatio. Page 1
2 CHAPTER 7: Cetral Limit Theorem: CLT for Averages (Meas) Class examples selected from those below; some but ot all problems or parts will be doe i class. EXAMPLE 1. Biology: A biologist fids that the legths of adult fish i a species of fish he is studyig follow a ormal distributio with a mea of 20 iches ad a stadard deviatio of 2 iches. a. Fid the probability that a idividual adult fish is betwee 19 ad 21 iches log. b. Fid the probability that for a sample of 4 adult fish, the average legth is betwee 19 ad 21 iches c. Fid the probability that for a sample of 16 adult fish, the average legth is betwee 19 ad 21 iches. d. Sketch the graphs of the probability distributios for a, b, ad c o the same axes showig how the shape of the distributio chages as the sample size chages. EXAMPLE 2. Percetiles for sample meas: A biologist fids that the legths of adult fish i a species of fish he is studyig follow a ormal distributio with a mea of 20 iches ad a stadard deviatio of 2 iches. a. Fid the 80 th percetile of idividual adult fish legths ad write a setece iterpretig the 80 th percetile. b. Fid the 80 th percetile of average fish legths for samples of 16 adult fish ad complete the iterpretatio. Iterpretatio: If we were to take repeated samples of 16 fish, 80% of all possible samples of 16 fish would have average legths of less tha iches. Page 2
3 CHAPTER 7: Cetral Limit Theorem: CLT for Averages (Meas) Class examples selected from those below; some but ot all problems or parts will be doe i class. EXAMPLE 3. Cetral Limit Theorem with Expoetial Distributio: Emergecy services such as "911" moitor the time iterval betwee calls received. Suppose that i a city, the time iterval betwee calls to "911" has a expoetial distributio, with a average of 5 miutes. a. Fid the probability that the time iterval util the ext call is betwee 4 ad 6 miutes. b. Fid the probability that the sample average time iterval is betwee 4 ad 6 miutes, for sample size = 36. c. Fid the probability that the sample average time iterval is betwee 4 ad 6 miutes, for sample size = 64. d. Sketch the graphs of the probability distributios for a, b, ad c o the same axes showig how the shape of the distributio chages as the sample size chages. EXAMPLE 4. Suppose that i a certai city, the time iterval betwee "911" calls has a expoetial distributio, with a average of 5 miutes. a. Fid the probability that the time iterval util the ext call is less tha 3 miutes. b. Fid the probability that the sample average time iterval is less tha 3 miutes for sample size = 64. Roud the probability to 5 decimal places. Page 3
4 CHAPTER 7: Cetral Limit Theorem: CLT for Averages (Meas) Class examples selected from those below; some but ot all problems or parts will be doe i class. EXAMPLE 5. Cetral Limit Theorem with Uiform Distributio: The ages of studets ridig school buses i a large city are uiformly distributed betwee 6 ad 16 years old. a. Fid the probability that oe radomly selected studet who rides the school bus is betwee 10 ad 12 years old. b. Fid the probability that the average age is betwee 10 ad 12 years for a radom sample of 30 studets who ride school buses. c. Sketch the graphs of the distributios for a ad b o oe set of axes. EXAMPLE 6. Evirometal Sciece: Power plats ad idustrial processes use water from sources such as rivers to regulate temperature. Water is take from the river, ru through pipes to cool the power or productio process, ad the released (clea) back ito the river. The temperature of released water is moitored closely. Fish ad plats livig i the river are very sesitive to the water temperature; small differeces ca affect survival. Suppose that water used to cool a idustrial process is released ito a river; the temperature of the released water has a ukow skewed right distributio with average temperature 14.1C ad stadard deviatio of 2.5C. a. Explai why you ca't fid the probability that the water released ito the river is more tha 15C b. Fid the probability that for a sample of 42 days, the average temperature of released water is more tha 15C. Page 4
5 CHAPTER 7: Cetral Limit Theorem: CLT for Averages (Meas) Coceptual Questios about the Cetral Limit Theorem Cetral Limit Theorem questios ca take the form of calculatio questios or cocept questios. As we did some of the calculatios i Examples 1 6, we discussed the cocepts. Try to aswer these questios that ask about uderstadig the coepts but do ot eed calculatios. EXAMPLE 7. Explai what happes to the mea of x as the sample size icreases. EXAMPLE 8a. Explai what happes to the stadard deviatio of x as the sample size icreases. 8b. Explai how this shows i the graph of the sample averages x as the sample size icreases. EXAMPLE 9. What happes to the shape of the distributio whe you look at sample averages istead of idividuals? EXAMPLE 10. Refers to Example 6 (power plats): Suppose that water used to cool a idustrial process is released ito a river; the temperature of the released water follows a ukow distributio that is skewed to the right with a average temperature of 14.1C with a stadard deviatio of 2.5C. For samples of 60 days, would the shape of the probability distributio for sample averages be skewed to the right, like the origial distributio of temperatures o idividual days? Explai why or why ot. How large does the sample size eed to be i order to use the Cetral Limit Theorem? The value of eeded to be a "large eough" sample size depeds o the shape of the origial distributio of the idividuals i the populatio If the idividuals i the origial populatio follow a ormal distributio, the the sample averages will have a ormal distributio, o matter how small or large the sample size is. If the idividuals i the origial populatio ( X ) do ot follow a ormal distributio, the the sample averages x become more ormally distributed as the sample size grows larger. I this case the sample averages x do ot follow the same distributio as the origial populatio. The more skewed the origial distributio of idividual values, the larger the sample size eeded. If the origial distributio is symmetric, the sample size eeded ca be smaller. May statistics textbooks use the rule of thumb 30, cosiderig 30 as the miimum sample size to use the Cetral Limit Theorem. But i reality there is ot a uiversal miimum sample size that works for all distributios; the sample size eeded depeds o the shape of the origial distributio. I your homework i chapter 7, assume the sample size is large eough for the Cetral Limit Theorem to be used to fid probabilities for x. Page 5
6 CHAPTER 7: Cetral Limit Theorem for Proportios The shape of the biomial distributio begis to approach a ormal distributio as the sample size gets larger, with a mea of = p ad pq Cosider a ifiite ( or very large) populatio that ca be divided ito two categories: success (the thig we are iterested i coutig) failure (aythig that is ot a success). For ay particular sample selected from this populatio, the sample proportio ca be calculated as: x umber of successes i the sample p pˆ umber of items or idividuals i the sample The symbols for the sample proportio are: p, said as p prime ; or pˆ, said as p hat. They both refer to the same thig: the proportio for a sample (rather tha for the whole populatio). The sample proportios are differet for differet samples due to radomess. The sample proportios from all the possible samples form a probability distributio, sice some values for the sample proportio are more likely to occur tha others. This probability distributio is called the samplig distributio for proportios. Cetral Limit Theorem for Proportios Let p be the probability of success, q be the probability of failure i the populatio x umber of successes i the sample p pˆ sample proportio umber of items or idividuals i the sample For sufficietly large samples of size draw from a ifiite or a very large populatio compared to the sample size, the the sample proportio p or pˆ is approximately ormally distributed with mea μ pˆ pˆ ~ N p, pq p ad stadard deviatio p ~ N p, pq σ pˆ We ll use this fact whe usig data from a sample to estimate a proportio for a populatio i chapter 8. (Techical Note: If the populatio is ot very large i compariso to the sample size the if we are samplig without replacemet we would eed use a slightly differet ormal distributio ivolvig a fiite populatio correctio factor whe calculatig the stadard deviatio.) EXAMPLE 11: Suppose that at a college, radom samples of 100 studets are selected. For each sample we determie the proportio of studets i the sample who receive some type of fiacial aid. Each sample has its ow sample proportio that may or may ot be the same as the populatio proportio. What is the probability distributio for the sample proportios? pq EXAMPLE 12: Toy s Pizza & Pasta Place sells 900 pizzas every moth; 40% of pizzas sold are delivered to customers ad 60% are either eate or picked up i the store by the customer. What is the probability distributio for the sample proportio of pizzas that are delivered to customers, for samples of = 900 pizzas. Page 6
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