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MATH225 Week 5 Assignment: Central Limit Theorem for Means
- Question A family of statisticians is trying to decide if they can afford for their child to play youth baseball. The cost of joining a team is normally distributed with a mean of $750 and a standard deviation of $185. If a sample of 40 teams is selected at random from the population, select the expected mean and standard deviation of the sampling distribution below. Correct answer: σx ¯=$29. μx ¯=$ The standard deviation of the sampling distribution σx ¯= σn −−√=$18540−−√=$29. When the distribution is normal the mean of the sampling distribution is equal to the mean of the population μx ¯= μ =$. Question A cupcake baker is planning a supplies order and needs to know how much flour he needs. He knows that his recipes use an average of 100 grams of flour, normally distributed, with a population standard deviation of 15 grams. If he is consulting a sample size of 30 recipes, select the mean and standard deviation of the sampling distribution to help him order his supplies from the options below. σx ¯=2.74 grams μx ¯=100 grams The standard deviation of the sampling distribution is σx ¯= σn −−√=1530−−√=2.74 grams
Likewise, when the distribution is normal the mean of the sampling distribution is equal to the mean of the population μx ¯= μ =100 grams. Question A head librarian for a large city is looking at the overdue fees per user system-wide to determine if the library should extend its lending period. The average library user has $19.67 in fees, with a standard deviation of $7.02. The data is normally distributed and a sample of 72 library users is selected at random from the population. Select the expected mean and standard deviation of the sampling distribution from the options below. Correct answer: σx ¯=$0. μx ¯=$19. The standard deviation of the sampling distribution is σx ¯= σn −−√=$7.0272−−√=$0. When the distribution is normal, the mean of the sampling distribution is equal to the mean of the population μx ¯= μ =$19..
- Question A well known social media company is looking to expand their online presence by creating another platform. They know that they current average 2,500,000 users each day, with a standard deviation of 625,000 users. If they randomly sample 50 days to analyze the use of their existing technology, identify each of the following, rounding to the nearest whole number if necessary:
To find the Standard Deviation of the sampling distribution, we divide the population standard deviation by the square root of the sample size: σx ¯= σn −−√=$24,500150−−−√=$2,000.
- Question The average time it takes a certain brand of ibuprofen to start working is 25 minutes, with a standard deviation of 13 minutes, distributed normally. A pharmacist randomly samples 20 pills from this brand, because she is researching different brands in order to find the quickest acting ibuprofen to recommend to her customers. Identify the following to help her make her recommendations, rounding to the nearest hundredth if necessary: We are given that the population mean is μ =25 minutes and that the population standard deviation σ =13 minutes, distributed normally. We want to find the mean and standard error of the sampling distribution, μx ¯ and σx ¯ for samples of size n =. By the Central Limit Theorem, the means of the two distributions are the same: μx ¯= μ =25 minutes To find the standard deviation of the sampling distribution, we divide the population standard deviation by the square root of the sample size: σx ¯= σn −−√=1320−−√=2.91 minutes
- Question Major league baseball recruiters are analyzing college players as potential draft choices. In a survey of college baseball players, the recruiters found that they hit an average of 13 home runs per season, with a standard deviation of 5. Suppose a random sample of 45 baseball players is selected. Identify each of the following and remember to round to the nearest whole number:
We are given population mean μ =13 and population standard deviation σ =5 , and want to find the mean and standard error of the sampling distribution, μx ¯ and σx ¯ for samples of size n =. By the Central Limit Theorem, the means of the two distributions are the same: μx ¯= μ = To find the Standard Deviation of the sampling distribution, we divide the population standard deviation by the square root of the sample size: σx ¯= σn −−√=545−−√≈
- Question The average credit card debt owed by Americans is $6375 , with a standard deviation of $1200. Suppose a random sample of 36 Americans is selected. Identify each of the following: We are given population mean μ =6375 and population standard deviation σ =1200 , and want to find the mean and standard error of the sampling distribution, μx ¯ and σx ¯ for samples of size n =. By the Central Limit Theorem, the means of the two distributions are the same: μx ¯= μ = To find the Standard Deviation of the sampling distribution, we divide the population standard deviation by the square root of the sample size: σx ¯= σn −−√=120036−−√=
- Question The heights of all basketball players are normally distributed with a mean of 72 inches and a population standard deviation of 1.5 inches. If a sample of 15 players
takes a random sample of 35 years to create a statistical study. Identify each of the following, rounding to the nearest hundredth when necessary: We are given population mean μ =150 and population standard deviation σ =18 , and want to find the mean and standard error of the sampling distribution, μx ¯ and σx ¯ for samples of size n =. By the Central Limit Theorem, the means of the two distributions are the same: μx ¯= μ = To find the Standard Deviation of the sampling distribution, we divide the population standard deviation by the square root of the sample size: σx ¯= σn −−√=18/35−−√≈3.
- Question Given the plot of normal distributions A and B below, which of the following statements is true? Select all correct answers. A curve labeled A rises to a maximum near the left of the horizontal axis and the falls. Another curve labeled B rises to a maximum to the right of and below curve A and falls. Correct answer: B has the larger mean.
B has the larger standard deviation. Remember that the mean of a normal distribution is the x -value of its central point (the top of the "hill"). Therefore, a distribution with a larger mean will be centered farther to the right than a distribution with a smaller mean. Because B is farther to the right than A , the mean of B is greater than the mean of A. Remember that the standard deviation tells how spread out the normal distribution is. So a high standard deviation means the graph will be short and spread out. A low standard deviation means the graph will be tall and skinny. Because B is shorter and more spread out than A , we find that B has the larger standard deviation.
- Question The graph below shows the graphs of several normal distributions, labeled A , B , and C , on the same axis. Determine which normal distribution has the largest standard deviation. A figure consists of three curves along a horizontal axis, labeled Upper A, Upper B and Upper C. Curve Upper A is farthest to the right, curve Upper B is tall and skinny, and curve Upper C is farthest to the left.
Because A and B are centered at the same point, their means are equal. Remember that the standard deviation tells how spread out the normal distribution is. So a high standard deviation means the graph will be short and spread out. A low standard deviation means the graph will be tall and skinny. Because B is shorter and more spread out than A , we find that B has the larger standard deviation.
- Which of the following lists of data has the smallest standard deviation? 12 , 12 , 8 , 12 , 11 , 12 , 12 , 9 , 11 , 12
- Which of the following lists of data has the smallest standard deviation? 17 , 19 , 17 , 18 , 17 , 16 , 16 , 16 , 17 , 20
- Question Given the plot of normal distributions A and B below, which of the following statements is true? Select all correct answers.
A figure consists of two curves labeled Upper A and Upper B. Curve Upper A is shorter and more spread out than curve Upper B, and the curve Upper B is taller and farther to the right than curve Upper A. Correct answer: B has the larger mean. A has the larger standard deviation. Remember that the mean of a normal distribution is the x -value of its central point (the top of the "hill"). Therefore, a distribution with a larger mean will be centered farther to the right than a distribution with a smaller mean. Because B is farther to the right than A , the mean of B is greater than the mean of A. Remember that the standard deviation tells how spread out the normal distribution is. So a high standard deviation means the graph will be short and spread out. A low standard deviation means the graph will be tall and skinny. Because A is shorter and more spread out than B , we find that A has the larger standard deviation.
- Question
A figure consists of three curves along a horizontal axis, labeled Upper A, Upper B and Upper C. Curve Upper A is farthest to the left from the center, curve Upper B is evenly spread out to the right from the center, and curve Upper C is tall and the least spread out. C Remember that the standard deviation tells how spread out the normal distribution is. So a high standard deviation means the graph will be short and spread out. A low standard deviation means the graph will be tall and skinny. The distribution that is the tallest and least spread out is C , so that has the smallest standard deviation.
- Question The graph below shows the graphs of several normal distributions, labeled A , B , and C , on the same axis. Determine which normal distribution has the smallest mean.
A curve labeled B rises to a maximum and then falls. A curve labeled A rises to a maximum below and to the right of A and then falls. A curve labeled C rises to a maximum to the right of and below the maximum of A. B Remember that the mean of a normal distribution is the x -value of its central point (the top of the "hill"). Therefore, a distribution with a larger mean will be centered farther to the right than a distribution with a smaller mean. The distribution that is farthest to the left is B , so that has the smallest mean.
- Question The graph below shows the graphs of several normal distributions, labeled A , B , and C , on the same axis. Determine which normal distribution has the smallest mean.
We are given population mean μ =170 and population standard deviation σ =45 , and want to find the mean and standard error of the sampling distribution, μx ¯ and σx ¯ for samples of size n =. By the Central Limit Theorem, the means of the two distributions are the same: μx ¯= μ = To find the Standard Deviation of the sampling distribution, we divide the population standard deviation by the square root of the sample size: σx ¯= σn −−√=45/31−−√≈
