MATH 225N Week 5 Assignment: Central Limit Theorem for Means.
1. Question
A family of statisticians is trying to decide if they can afford for their
... [Show More] 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¯=$29.25
μx¯=$750
The standard deviation of the sampling distribution
σx¯=σn−−√=$18540−−√=$29.25
When the distribution is normal the mean of the sampling distribution is equal to the mean of the population μx¯=μ=$750 .
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¯=$0.83
μx¯=$19.67
The standard deviation of the sampling distribution is
σx¯=σn−−√=$7.0272−−√=$0.83
When the distribution is normal, the mean of the sampling distribution is equal to the mean of the population μx¯=μ=$19.67 .
2. 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:
We are given population mean μ=2,500,000 and population standard deviation σ=625,000 , and want to find the mean and standard error of the sampling distribution, μx¯ and σx¯ for samples of size n=50 .
By the Central Limit Theorem, the means of the two distributions are the same:
μx¯=μ=2,500,000
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−−√=625,00050−−√≈88,388
3. Question
A bank is reviewing its risk management policies with regards to mortgages. To minimize the risk of lending, the bank wants to compare the typical mortgage owed by their clients against other homebuyers. The average mortgage owed by Americans is $306,500 , with a standard deviation of $24,500 . Suppose a random sample of 150 Americans is selected.
Identify each of the following, rounding your answers to the nearest cent when appropriate:
• 1306500$306500$306500
• 224500$24500$24500
• 3150$150$150
• 4306500$306500$306500
• $2000.42
We are given population mean μ=$306,500 and population standard deviation σ=$24,500 , and want to find the mean and standard error of the sampling distribution, μx¯ and σx¯ for samples of size n=150 .
By the Central Limit Theorem, the means of the two distributions are the same:
μx¯=μ=$306,500
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−−−√=$2,000.42
4. 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=20 .
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
5. 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=45 .
By the Central Limit Theorem, the means of the two distributions are the same:
μx¯=μ=13
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−−√≈1
6. 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=36 .
By the Central Limit Theorem, the means of the two distributions are the same:
μx¯=μ=6375
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−−√=200
7. 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 are selected at random from the population, select the expected mean of the sampling distribution and the standard deviation of the sampling distribution below.
Correct answer:
σx¯=0.387 inches
μx¯=72 inches
The standard deviation of the sampling distribution σx¯=σn√=1.51√5=0.387 inches. Likewise, when the distribution is normal the mean of the sampling distribution is equal to the mean of the population μx¯=μ=72 inches
8. Question
The owners of a baseball team are building a new baseball field for their team and must determine the number of seats to include. The average game is attended by 6,500 fans, with a standard deviation of 450 people. Suppose a random sample of 10 games is selected to help the owners decide the number of seats to include. Identify each of the following and be sure to round to the nearest whole number:
We are given population mean μ=6,500 and population standard deviation σ=450 , and want to find the mean and standard error of the sampling distribution, μx¯ and σx¯ for samples of size n=10 .
By the Central Limit Theorem, the means of the two distributions are the same:
μx¯=μ=6,500
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−−√=45010−−√≈142
9. Question
The Washington Wheat Farmers Club is studying the impact of rising grain prices on their members' planting habits. The club members produce an average of 150 million bushels of wheat per year, with a standard deviation of 18 million bushels. The club 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=36 .
By the Central Limit Theorem, the means of the two distributions are the same:
μx¯=μ=150
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−−√≈3.04
10. 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.
11. 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.
Correct answer:
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 most spread out is C, so that has the largest standard deviation.
12. 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. The curve Upper A is tall and evenly spread out from the center and the curve Upper is B is shorter and more spread out than A.
Correct answer:
The means of A and B are equal.
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 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.
13. Which of the following lists of data has the smallest standard deviation?
________________________________________
________________________________________
12, 12, 8, 12, 11, 12, 12, 9, 11, 12
14. Which of the following lists of data has the smallest standard deviation?
17 , 19 , 17 , 18 , 17 , 16 , 16 , 16 , 17 , 20
15. 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.
16. 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 standard deviation.
A figure consists of three curves along a horizontal axis, labeled Upper A, Upper B and Upper C. Curve Upper A is evenly spread out, curve Upper B is tall and the least spread out, and curve Upper C is short and more evenly spread out from the center.
B
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 B, so that has the smallest standard deviation.
17. 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 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 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.
18. 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.
19. 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 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, curve Upper B is farthest to the right, and curve Upper C is tall and skinny.
________________________________________
A
20. A head football coach is concerned about the weight gain of some of his players. He finds that the weight of all football players is normally distributed with a mean of 250 pounds and a population standard deviation of 54 pounds. If the coach selects a random sample of 10 players from the population, identify the expected mean and the standard deviation of the sampling distribution below.
σx¯=17.08 pounds
μx¯=250 pounds
21. Question
A businesswoman wants to open a coffee stand across the street from a competing coffee company. She notices that the competing company has an average of 170 customers each day, with a standard deviation of 45 customers. Suppose she takes a random sample of 31 days. Identify the following to help her decide whether to open her coffee stand, rounding to the nearest whole number when necessary:
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=31 .
By the Central Limit Theorem, the means of the two distributions are the same:
μx¯=μ=170
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−−√≈8
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