Building Better Batteries
Everyone wants to have the latest technological gadget. That’s why iPods, digital cameras, PDAs, Game Boys, and camera phones have solid millions of units. These devices require lots of power and can drain traditional alkaline batteries quickly. Battery manufacturers are constantly searching for ways to build longer-lasting batteries. In July 2005, Panasonic began marketing its new Oxyride battery in the United States. According to the results of preliminary testing, Oxyride batteries produced more power and lasted up to twice as long as alkaline batteries.
Battery manufacturers must constantly measure battery lifetimes to ensure that their production process is working properly. Because testing a battery’s lifetime requires the battery to be drained completely, the manufacturer wants to test as few batteries as possible. As part of the quality control process, the manufacturer selects a sample of batteries to test at regular intervals throughout production. By looking at the results from the sample, the manufacturer can determine whether the entire batch of batteries produced meets specifications.
At a particular battery production plant, when the process is working properly, AA batteries last an average of 17 hours with a standard deviation of 0.8 hour. Quality control inspectors select a random sample of 30 batteries during each hour of production and then drain them under conditions that mimic normal use. Here are the lifetimes (in hours) of the batteries from one such sample:
16.91 18.83 17.58 15.84 17.42 17.65 16.63 16.84 15.63 16.37
15.80 15.93 15.81 17.45 16.85 16.33 16.22 16.59 17.13 17.10
16.96 16.40 17.35 16.37 15.98 16.52 17.04 17.07 15.73 16.74
Do theses data suggest that the process is working properly?
1. Assuming the process is working properly (mean = 17, standard deviation =0.8), what are the shape, center, and spread of the distribution of sample means for random samples of 30 batteries? Justify each of your answers.
2. For the random sample of 30 batteries, =16.70 hours. Calculate the probability of obtaining a sample with a mean lifetime of 16.70 hours or less if the production process is working properly. Show your work.
3. Based on your answer to Question 2, do you believe that the process is working properly? Justify your answer.
The plant manager also wants to know what proportion of all the batteries produced that day lasted less than 16 hours, which he has declared "unsuitable." From past experience, about 10% of batteries made at the plant are unsuitable. If he can feel relativley certain that the proportion of unsuitable batteries "p" produced that day is less that 0.10, he might take the chance of shipping the whole batch of batteries to customers.
4. Assuming that the actual proportion of unsuitable batteries produced that day is 0.10, what are the shape, center, and spread of the distribution of sample porportions for random samples of 480 batteries? Justify each of your answers.
5. On this particular day, a total of 480 batteries were tested during 16 hours of production. Forty of these batteries were unsuitable. Calculate the probability of obtaining a sample in which such a samll proportion of batteries is unsuitable if p = 0.10. Show your work.
6. Based on your answer to question 5, what advice would you give the plant manager? Why?
Resources:
www.stat.sc.edu/~west/javahtml/CLT.html
Beth Chance's Reese's Pieces applet and sampling pennies www.rossmanchance.com/applets/index.html
David Lane's Sampling Distribution
www.ruf.rice.edu/~lane/stat_sim/sampling_dist/
Central Limit Theorem
www.whfreeman.com/tps3e
Friday, February 12, 2010
AP Statistics Group 2 (Period 7)
Baggage Check
Thousands of travelers pass through Guadalajara airport each day. Before leaving the airport, each passenger must pass through the Customs inspection area. Customs officials want to be sure that passengers do not bring illegal items into the country. But they do not have time to search every traveler’s luggage. Instead, they require each person to press a button. Either a red or a green bulb lights up. If the red light shows, the passenger will be searched by Customs agents. A green light means “go ahead.” Customs officers claim that the probability that the light turns green on any press of the button is 0.70. You observe 100 passengers at the Customs “stop-light.” Only 65 get a green light. Does this give you reason to doubt the Customs officials?
a.) Use your calculator to simulate 50 groups of 100 passengers activating the Customs light. Use the command randBin(100, 0.7, 50)/100 store in L1.List 1 will contain 50 values of , the proportion of the 100 passengers who got a green light.
b.) Sort L 1 in descending order. In how many of the 50 simulations did you obtain a value of that is less than or equal to 0.65? Do you believe the Customs agents?
c.) Describe the shape, center, and spread of the sampling distribution of for samples of n = 100 passengers.
d.) Use the sampling distribution from part (c) to find the probability of getting a sample proportion of 0.65 or less if p=0.7 is actually true. How does this compare with the results of your simulation in part (b)?
e.) Repeat parts (c) and (d) for samples of size n = 1000 passengers. Report your findings as compared to n = 100.
Resources:
Beth Chance's Reese's Pieces applet and sampling pennies www.rossmanchance.com/applets/index.html
David Lane's Sampling Distribution
www.ruf.rice.edu/~lane/stat_sim/sampling_dist/
Central Limit Theorem
www.whfreeman.com/tps3e
Thousands of travelers pass through Guadalajara airport each day. Before leaving the airport, each passenger must pass through the Customs inspection area. Customs officials want to be sure that passengers do not bring illegal items into the country. But they do not have time to search every traveler’s luggage. Instead, they require each person to press a button. Either a red or a green bulb lights up. If the red light shows, the passenger will be searched by Customs agents. A green light means “go ahead.” Customs officers claim that the probability that the light turns green on any press of the button is 0.70. You observe 100 passengers at the Customs “stop-light.” Only 65 get a green light. Does this give you reason to doubt the Customs officials?
a.) Use your calculator to simulate 50 groups of 100 passengers activating the Customs light. Use the command randBin(100, 0.7, 50)/100 store in L1.List 1 will contain 50 values of , the proportion of the 100 passengers who got a green light.
b.) Sort L 1 in descending order. In how many of the 50 simulations did you obtain a value of that is less than or equal to 0.65? Do you believe the Customs agents?
c.) Describe the shape, center, and spread of the sampling distribution of for samples of n = 100 passengers.
d.) Use the sampling distribution from part (c) to find the probability of getting a sample proportion of 0.65 or less if p=0.7 is actually true. How does this compare with the results of your simulation in part (b)?
e.) Repeat parts (c) and (d) for samples of size n = 1000 passengers. Report your findings as compared to n = 100.
Resources:
Beth Chance's Reese's Pieces applet and sampling pennies www.rossmanchance.com/applets/index.html
David Lane's Sampling Distribution
www.ruf.rice.edu/~lane/stat_sim/sampling_dist/
Central Limit Theorem
www.whfreeman.com/tps3e
Thursday, February 11, 2010
AP Statistics Group 2 (Period 3)
Baggage Check
Thousands of travelers pass through Guadalajara airport each day. Before leaving the airport, each passenger must pass through the Customs inspection area. Customs officials want to be sure that passengers do not bring illegal items into the country. But they do not have time to search every traveler’s luggage. Instead, they require each person to press a button. Either a red or a green bulb lights up. If the red light shows, the passenger will be searched by Customs agents. A green light means “go ahead.” Customs officers claim that the probability that the light turns green on any press of the button is 0.70. You observe 100 passengers at the Customs “stop-light.” Only 65 get a green light. Does this give you reason to doubt the Customs officials?
a.) Use your calculator to simulate 50 groups of 100 passengers activating the Customs light. Use the command randBin(100, 0.7, 50)/100 store in L1.List 1 will contain 50 values of , the proportion of the 100 passengers who got a green light.
b.) Sort L 1 in descending order. In how many of the 50 simulations did you obtain a value of that is less than or equal to 0.65? Do you believe the Customs agents?
c.) Describe the shape, center, and spread of the sampling distribution of for samples of n = 100 passengers.
d.) Use the sampling distribution from part (c) to find the probability of getting a sample proportion of 0.65 or less if p=0.7 is actually true. How does this compare with the results of your simulation in part (b)?
e.) Repeat parts (c) and (d) for samples of size n = 1000 passengers. Report your findings as compared to n = 100.
Resources:
Beth Chance's Reese's Pieces applet and sampling pennies www.rossmanchance.com/applets/index.html
David Lane's Sampling Distribution
www.ruf.rice.edu/~lane/stat_sim/sampling_dist/
Central Limit Theorem
www.whfreeman.com/tps3e
Thousands of travelers pass through Guadalajara airport each day. Before leaving the airport, each passenger must pass through the Customs inspection area. Customs officials want to be sure that passengers do not bring illegal items into the country. But they do not have time to search every traveler’s luggage. Instead, they require each person to press a button. Either a red or a green bulb lights up. If the red light shows, the passenger will be searched by Customs agents. A green light means “go ahead.” Customs officers claim that the probability that the light turns green on any press of the button is 0.70. You observe 100 passengers at the Customs “stop-light.” Only 65 get a green light. Does this give you reason to doubt the Customs officials?
a.) Use your calculator to simulate 50 groups of 100 passengers activating the Customs light. Use the command randBin(100, 0.7, 50)/100 store in L1.List 1 will contain 50 values of , the proportion of the 100 passengers who got a green light.
b.) Sort L 1 in descending order. In how many of the 50 simulations did you obtain a value of that is less than or equal to 0.65? Do you believe the Customs agents?
c.) Describe the shape, center, and spread of the sampling distribution of for samples of n = 100 passengers.
d.) Use the sampling distribution from part (c) to find the probability of getting a sample proportion of 0.65 or less if p=0.7 is actually true. How does this compare with the results of your simulation in part (b)?
e.) Repeat parts (c) and (d) for samples of size n = 1000 passengers. Report your findings as compared to n = 100.
Resources:
Beth Chance's Reese's Pieces applet and sampling pennies www.rossmanchance.com/applets/index.html
David Lane's Sampling Distribution
www.ruf.rice.edu/~lane/stat_sim/sampling_dist/
Central Limit Theorem
www.whfreeman.com/tps3e
AP Statistics Group 1 (Period 3) Courtney A, kemp, bowden, young,cifarelli, woodhouse, decaprio,wheeler, derosa,vrabac, dinuzzo, tran,greco
Building Better Batteries
Everyone wants to have the latest technological gadget. That’s why iPods, digital cameras, PDAs, Game Boys, and camera phones have solid millions of units. These devices require lots of power and can drain traditional alkaline batteries quickly. Battery manufacturers are constantly searching for ways to build longer-lasting batteries. In July 2005, Panasonic began marketing its new Oxyride battery in the United States. According to the results of preliminary testing, Oxyride batteries produced more power and lasted up to twice as long as alkaline batteries.
Battery manufacturers must constantly measure battery lifetimes to ensure that their production process is working properly. Because testing a battery’s lifetime requires the battery to be drained completely, the manufacturer wants to test as few batteries as possible. As part of the quality control process, the manufacturer selects a sample of batteries to test at regular intervals throughout production. By looking at the results from the sample, the manufacturer can determine whether the entire batch of batteries produced meets specifications.
At a particular battery production plant, when the process is working properly, AA batteries last an average of 17 hours with a standard deviation of 0.8 hour. Quality control inspectors select a random sample of 30 batteries during each hour of production and then drain them under conditions that mimic normal use. Here are the lifetimes (in hours) of the batteries from one such sample:
16.91 18.83 17.58 15.84 17.42 17.65 16.63 16.84 15.63 16.37
15.80 15.93 15.81 17.45 16.85 16.33 16.22 16.59 17.13 17.10
16.96 16.40 17.35 16.37 15.98 16.52 17.04 17.07 15.73 16.74
Do theses data suggest that the process is working properly?
1. Assuming the process is working properly (mean = 17, standard deviation =0.8), what are the shape, center, and spread of the distribution of sample means for random samples of 30 batteries? Justify each of your answers.
2. For the random sample of 30 batteries, =16.70 hours. Calculate the probability of obtaining a sample with a mean lifetime of 16.70 hours or less if the production process is working properly. Show your work.
3. Based on your answer to Question 2, do you believe that the process is working properly? Justify your answer.
The plant manager also wants to know what proportion of all the batteries produced that day lasted less than 16 hours, which he has declared "unsuitable." From past experience, about 10% of batteries made at the plant are unsuitable. If he can feel relativley certain that the proportion of unsuitable batteries "p" produced that day is less that 0.10, he might take the chance of shipping the whole batch of batteries to customers.
4. Assuming that the actual proportion of unsuitable batteries produced that day is 0.10, what are the shape, center, and spread of the distribution of sample porportions for random samples of 480 batteries? Justify each of your answers.
5. On this particular day, a total of 480 batteries were tested during 16 hours of production. Forty of these batteries were unsuitable. Calculate the probability of obtaining a sample in which such a samll proportion of batteries is unsuitable if p = 0.10. Show your work.
6. Based on your answer to question 5, what advice would you give the plant manager? Why?
Resources:
www.stat.sc.edu/~west/javahtml/CLT.html
Beth Chance's Reese's Pieces applet and sampling pennies www.rossmanchance.com/applets/index.html
David Lane's Sampling Distribution
www.ruf.rice.edu/~lane/stat_sim/sampling_dist/
Central Limit Theorem
www.whfreeman.com/tps3e
Everyone wants to have the latest technological gadget. That’s why iPods, digital cameras, PDAs, Game Boys, and camera phones have solid millions of units. These devices require lots of power and can drain traditional alkaline batteries quickly. Battery manufacturers are constantly searching for ways to build longer-lasting batteries. In July 2005, Panasonic began marketing its new Oxyride battery in the United States. According to the results of preliminary testing, Oxyride batteries produced more power and lasted up to twice as long as alkaline batteries.
Battery manufacturers must constantly measure battery lifetimes to ensure that their production process is working properly. Because testing a battery’s lifetime requires the battery to be drained completely, the manufacturer wants to test as few batteries as possible. As part of the quality control process, the manufacturer selects a sample of batteries to test at regular intervals throughout production. By looking at the results from the sample, the manufacturer can determine whether the entire batch of batteries produced meets specifications.
At a particular battery production plant, when the process is working properly, AA batteries last an average of 17 hours with a standard deviation of 0.8 hour. Quality control inspectors select a random sample of 30 batteries during each hour of production and then drain them under conditions that mimic normal use. Here are the lifetimes (in hours) of the batteries from one such sample:
16.91 18.83 17.58 15.84 17.42 17.65 16.63 16.84 15.63 16.37
15.80 15.93 15.81 17.45 16.85 16.33 16.22 16.59 17.13 17.10
16.96 16.40 17.35 16.37 15.98 16.52 17.04 17.07 15.73 16.74
Do theses data suggest that the process is working properly?
1. Assuming the process is working properly (mean = 17, standard deviation =0.8), what are the shape, center, and spread of the distribution of sample means for random samples of 30 batteries? Justify each of your answers.
2. For the random sample of 30 batteries, =16.70 hours. Calculate the probability of obtaining a sample with a mean lifetime of 16.70 hours or less if the production process is working properly. Show your work.
3. Based on your answer to Question 2, do you believe that the process is working properly? Justify your answer.
The plant manager also wants to know what proportion of all the batteries produced that day lasted less than 16 hours, which he has declared "unsuitable." From past experience, about 10% of batteries made at the plant are unsuitable. If he can feel relativley certain that the proportion of unsuitable batteries "p" produced that day is less that 0.10, he might take the chance of shipping the whole batch of batteries to customers.
4. Assuming that the actual proportion of unsuitable batteries produced that day is 0.10, what are the shape, center, and spread of the distribution of sample porportions for random samples of 480 batteries? Justify each of your answers.
5. On this particular day, a total of 480 batteries were tested during 16 hours of production. Forty of these batteries were unsuitable. Calculate the probability of obtaining a sample in which such a samll proportion of batteries is unsuitable if p = 0.10. Show your work.
6. Based on your answer to question 5, what advice would you give the plant manager? Why?
Resources:
www.stat.sc.edu/~west/javahtml/CLT.html
Beth Chance's Reese's Pieces applet and sampling pennies www.rossmanchance.com/applets/index.html
David Lane's Sampling Distribution
www.ruf.rice.edu/~lane/stat_sim/sampling_dist/
Central Limit Theorem
www.whfreeman.com/tps3e
Algebra II Math and Careers
In this assignment you will research a career that you are interested in pursuing. You will have to research the required math classes that you will need to complete in order to obtain this career. (college websites, etc.) Give examples of ways that math may be used in your particular career choice. Explain how previous and current math courses have prepared you to reach your career goal.
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