Central Limit Theorem for Proportions Week 5 Assignment PDF

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Central Limit Theorem for Proportions Week 5 Assignment...

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Central Limit Theorem for Proportions Week 5 Assignment: 1. A dental student is conducting a study on the number of people who visit their dentist regularly. Of the 520 people surveyed, 312 indicated that they had visited their dentist within the past year. Find the population proportion, as well as the mean and standard deviation of the sampling distribution for samples of size n=60.

Correct answers: •

10 point 6 0 0 $0.600$0.600

•

20 point 6 0 0 $0.600$0.600

•

30 point 0 6 3 $0.063$0.063

The population proportion is p=312520≈0.600. This is also the mean of the sampling distribution:

μp^=p=0.600 For samples of size n=60, the standard deviation of the sampling distribution is

σpˆ=p(1−p)n−−−−−−−√=0.600(1−0.600)60−−−−−−−−−−−−−−√≈0.063 2. From recent census data, it is discovered that the proportion of the adults in the United States who are first generation Americans is 14%. For a random sample of size 500, what is standard deviation for the sampling distribution of the sample proportions, rounded to three decimal places?

---Given the population proportion p=14%=0.14 and a sample size of n=500, the standard deviation of the sampling distribution of sample proportions is

σpˆ=p(1−p)n−−−−−−−√=0.14(1−0.14)500−−−−−−−−−−−−√≈0.016

3. From recent survey data, a car buying business finds that 12.5% of the proportion of adults in a city would be likely to use their services. For a random sample of 115 people, what is standard deviation for the sampling distribution of the sample proportions, rounded to three decimal places?

Given the population proportion p=12.5%=0.125 and a sample size of n=115, the standard deviation of the sampling distribution of sample proportions is

σpˆ=p(1−p)n−−−−−−−√=0.125(1−0.125)115−−−−−−−−−−−−−−√≈0.031 4. Question From recent survey data, its known that the proportion of adults in the United States who are smokers is 18%. For a random sample of size 150, what is standard deviation for the sampling distribution of the sample proportions, rounded to three decimal places? Given the population proportion p=18%=0.18 and a sample size of n=150, the standard deviation of the sampling distribution of sample proportions is

σpˆ=p(1−p)n−−−−−−−√=0.18(1−0.18)150−−−−−−−−−−−−√≈0.031 5. A small community college has a population of 986 students, 252 of whom them are considered non-traditional students. Find the population proportion, as well as the mean and standard deviation of the sampling distribution for samples of size n=40. Round all answers to 3 decimal places. The population proportion is p=252986≈0.256. This is also the mean of the sampling distribution:

μp^=p=0.256 For samples of size n=40, the standard deviation of the sampling distribution is

σpˆ=p(1−p)n−−−−−−−√=0.256(1−0.256)40−−−−−−−−−−−−−−√≈0.069 6. A marketing team is targeting people who might buy a hybrid car. In their city, with a

population of 30,000 people, 3,170 people either drive a hybrid car or have indicated on a recent survey that they would be interested in driving one. Find the population proportion, as well as the mean and standard deviation of the sampling distribution for samples of size n=121.

he population proportion is p=3,17030,000≈0.106. This is also the mean of the sampling distribution:

μp^=p=0.106 For samples of size n=121, the standard deviation of the sampling distribution is

σpˆ=p(1−p)n−−−−−−−√=0.106(1−0.106)121−−−−−−−−−−−−−−√≈0.028 7. A brokerage company is analyzing its retirement accounts for its clients nearing retirement. From recent survey data, the proportion of adults in the United States who say they are financially ready for retirement is 31%. For a random sample of size 85, what is standard deviation for the sampling distribution of the sample proportions, rounded to three decimal places?

Given the population proportion p=31%=0.31 and a sample size of n=85, the standard deviation of the sampling distribution of sample proportions is

σpˆ=p(1−p)n−−−−−−−√=0.31(1−0.31)85−−−−−−−−−−−−√≈0.050 8. Question Given the plot of normal distributions A and B below, which of the following statements is true? Select all correct answers.

B has the larger mean. A has the larger standard deviation. 9.

Question

Given the plot of normal distributions A and B below, which of the following statements is true? Select all correct answers.

The means of A and B are equal.

A has the larger standard deviation. 10.

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.

11.

Question

A

Given the plot of normal distributions A and B below, which of the following statements is true? Select all correct answers.

B has the larger mean. A has the larger standard deviation. 12.

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 mean.

A 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 right is A, so that has the largest mean. 13. Question Which of the data sets represented by the following histograms has the smallest standard deviation?

An untitled histogram has a horizontal axis labeled from 0 to 150 in increments of 50 and vertical axis labeled from 0 to 140 in increments of 20. The histogram contains vertical bars of width 6, starting at the horizontal axis value of 75. The approximate heights of the bars as follows, where the horizontal axis label is listed first and the approximate height is listed second: 75, 6; 82, 18; 88, 70; 76, 115; 82, 150; 88, 103; 95, 70; 102, 20; 108, 8. Remember that the standard deviation is a measure of how spread out the data is. If the values are concentrated around the mean, then a data set has a lower standard deviation.

A histogram with a taller hill indicates higher frequencies of the same values, and has a lower standard deviation than a histogram with a shorter hill and a wider range of values. 14. Question Which of the data sets represented by the following box and whisker plots has the smallest standard deviation?

B

15. Suppose that you are conducting a survey on how many pets each employee has in his or her household. The mean number of pets is 4 per household, and the standard deviation is

2. Rob only owns cats, and he has 10 cats. (Which of the following statements is true?) Correct answer: The number of pets owned by Rob is 3 standard deviations to the right of the mean. Mean=4. SD=2.

If the number of pets owned by Rob is 6 standard deviations to the right of the mean, then Rob should have 4+6×2=16 pets. But Rob has =10 pets.

16. Question A math instructor is teaching an algebra course, statistics course and calculus course. After the final exams are completed the instructor would like to know which class had most consistent scores on the final exam and which class had the greatest variation in scores on the final exam. The results from the final exam for these three classes are shown below: • • •

Algebra class: Mean Final Exam score of 85.2, standard deviation of 3.9 Statistics class: Mean Final Exam score of 81.7, standard deviation of 5.3 Calculus class: Mean Final Exam score of 84.3, standard deviation of 1.8

Based on these results, which class had most consistent scores on the final exam and which class had the greatest variation in scores on the final exam.

Calculus class had the most consistent scores, Statistics class had the greatest variation in scores.

17. Question Which of the data sets represented by the following box and whisker plots has the smallest standard deviation?

B

18. Which of the following lists of data has the largest standard deviation?

Answer Explanation Correct answer: 7 , 14 , 19 , 18 , 4 , 16 , 5 , 11 , 11 , 12

19. Question Which of the following lists of data has the largest standard deviation? 24, 15, 21, 23, 9, 22, 12, 21, 20, 13

20. 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 mean.

A

21.

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 evenly spread out from the center, and curve Upper B is farther to the right than A and more evenly spread out.

B has the larger mean. The standard deviations of A and B are equal. 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 the distributions are the same height and equally spread out, we find that the standard deviations of A and B are equal.

$$$Graphs

of the Normal Distribution

There are many normal distributions, and these normal distributions can differ in their means and in their standard deviations. The figure below shows three normal distributions.

A figure consists of three unlabeled curves. The first curve is evenly spread out from the center, the second curve is tall and the least spread out, and the third curve is taller and skinnier than the second curve, and it is the least spread out. • • •

The distribution in green (left-most) has a mean of −3 and a standard deviation of 0.5. The distribution in red (in the middle) has a mean of 0 and a standard deviation of 1. The distribution in black (right-most) has a mean of 2 and a standard deviation of 3.

These, as well as all other normal distributions, are symmetric with more values at the center of the distribution than in the tails. Note that the standard deviation affects the height of the normal distribution.

Parameters of the Normal Distribution The density of the normal distribution (the height for a given value on the x-axis) is a complicated expression.

f(x)=1σ2π−−√e−(x−μ)2/(2σ2)

The most important part of this expression are the parameters that define the normal distribution: the mean (μ) and the standard deviation (σ). The symbols e and π are mathematical constants. Seven features of normal distributions are listed below: 1. 2. 3. 4. 5.

Normal distributions are symmetric around their mean. The mean, median, and mode of a normal distribution are equal. The area under the normal curve is equal to 1. Normal distributions are denser in the center and less dense in the tails. Normal distributions are defined by two parameters, the mean (μ) and the standard deviation (σ).

68% of the area of a normal distribution is within one standard deviation of the mean. 7. Approximately 95% of the area of a normal distribution is within two standard deviations of the mean.

6.

22. 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.

C

23. Question

An obstetrician is researching postpartum programs for new mothers. From recent census data, it is known that the proportion of women in the United States who are of child bearing age is 17%. For a random sample of size 250, what is standard deviation for the sampling distribution of the sample proportions, rounded to three decimal places?

•

Correct answers: 0 point 0 2 4 $0.024$0.024 Given the population proportion p=17%=0.17 and a sample size of n=250, the standard deviation of the sampling distribution of sample proportions is

σpˆ=p(1−p)n−−−−−−−√=0.17(1−0.17)250−−−−−−−−−−−−√≈0.024 24. Question A youth director for a city's parks and recreations department is planning for next year's program. From a a survey of city residents, the director knows that 35% of the population participate in the parks and recreations programs. For a random sample of 40 people, what is standard deviation for the sampling distribution of the sample proportions, rounded to three decimal places? Given the population proportion p=35%=0.35 and a sample size of n=40, the standard deviation of the sampling distribution of sample proportions is

σpˆ=p(1−p)n−−−−−−−√=0.35(1−0.35)40−−−−−−−−−−−−√≈0.075 25. From recent survey data, its known that the proportion of adults in the United States who are smokers is 18%. For a random sample of size 150, what is standard deviation for the sampling distribution of the sample proportions.

Given the population proportion p=18%=0.18 and a sample size of n=150, the standard deviation of the sampling distribution of sample proportions is

σpˆ=p(1−p)n−−−−−−−√=0.18(1−0.18)150−−−−−−−−−−−−√≈0.031 26. A small community college has a population of 986 students, 252 of whom them are considered non-traditional students. Find the population proportion, as well as the mean and standard deviation of the sampling distribution for samples of size n=40.

The population proportion is p=252986≈0.256. This is also the mean of the sampling distribution:

μp^=p=0.256 For samples of size n=40, the standard deviation of the sampling distribution is

σpˆ=p(1−p)n−−−−−−−√=0.256(1−0.256)40−−−−−−−−−−−−−−√≈0.069

27. Economists are analyzing the effects of new tax policies on population growth. From recent census data, it is known that the proportion of families in the United States who have more than

3 children in the household is 26%. For a random sample of 50 families, what is the standard deviation for the sampling distribution of the sample proportions, rounded to three decimal places? Given the population proportion p=26%=0.26 and a sample size of n=50, the

standard deviation of the sampling distribution of sample proportions is

σpˆ=p(1−p)n−−−−−−−√=0.26(1−0.26)50−−−−−−−−−−−−√≈0.062 28. Question Of the 459,000 people living in Miami, 68%indicated on a recent, city-wide survey that they were employed . For a random sample of size 490 people, what is standard deviation for the sampling distribution of the sample proportions, rounded to three decimal places? Given the

population proportion p=68%=0.68 and a sample size of n=490, the standard deviation of the sampling distribution of sample proportions is

σpˆ=p(1−p)n−−−−−−−√=0.68(1−0.68)490−−−−−−−−−−−−√≈0.021 29. 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 A rises to a maximum near the left of the horizontal axis and falls, a curve labeled B rises to a maximum near the right of the horizontal axis and falls, a curve labeled C rises to a maximum near the center of the horizontal axis and falls. Correct answer:

A 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 A, so that has the smallest mean.

31. Question A credit card company surveys its customers to determine the number of times they use the card each month. There are 5,500 customers and 2,756 indicate that they use the card at least twice each month. Find the population proportion, as well as the mean and standard deviation of the sampling distribution for samples of size n=70 customers. The population proportion is p=2,7565,500≈0.501. This is also the mean of the sampling distribution:

μp^=p=0.501

For samples of size n=70, the standard deviation of the sampling distribution is

σpˆ=p(1−p)n−−−−−−−√=0.501(1−0.501)70−−−−−−−−−−−−−−√≈0.060 32. Question A horse racing track knows that the proportion of its gamblers that report winning is 15%. For a random sample of 64 gamblers, what is standard deviation for the sampling distribution of the sample proportions, rounded to three decimal places? Given the population proportion p=15%=0.15 and a sample size of n=64, the standard deviation of the sampling distribution of sample proportions is

σpˆ=p(1−p)n−−−−−−−√=0.15(1−0.15)64−−−−−−−−−−−−√≈0.045 33. Question Of the 13,500 savings accounts in a bank, 4,675 belong to people younger than 40 years old. The bank president would like to increase her institution's marketing strategy to younger customers, so she is examining the population proportions in order to create a statistical study. Find the population proportion, as well as the mean and standard deviation of the sampling distribution for samples of size n=400.

The population proportion is p=4,675/13,500≈0.346. This is also the mean of the sampling distribution:

μp^=p=0.346 For samples of size n=400, the standard deviation of the sampling distribution is

σpˆ=p(1−p)n−−−−−−−√=0.346(1−0.346)400−−−−−−−−−−−−−−√≈0.024 34. A baseball team calls itself "America's Favorite Team," because it has 90,000 fans on social media out of 2,210,000 social media users. Find the population proportion, as well as the mean and standard deviation of the sampling distribution for samples of size n=1,000. Round all answers to 3 decimal places.

The population proportion is p=90,000/2,210,000≈0.041. This is also the mean of the sampling distribution:

μp^=p=0.041 For samples of size n=1,000, the standard deviation of the sampling distribution is

σpˆ=p(1−p)n−−−−−−−√=0.041(1−0.041)1,000−−−−−−−−−−−−−−√≈0.006 35. Question A study by doctors of children attending a certain elementary school finds that, of 812 students, 245 watch more than 2 hours of television each day. Find the population proportion, as well as the mean and standard deviation of the sampling distribution for samples of size n=100 students. Round all answers to 3 decimal places. The population proportion is

p=245812≈0.302 . This is also the mean of the sampling

distribution:

μp^=p=0.302 For samples of size n=100 , the standard deviation of the sampling distribution is

σpˆ=p(1−p)n−−−−−−−√=0.302(1−0.302)100−−−−−−−−−−−−−−√≈0.046 36. Pharmacy technicians are concerned about the rising number of fraudulent prescriptions they are seeing. A small pharmacy sees 1,500 new prescriptions a month, 28 of which are fraudulent. Find the population proportion, as well as the mean and standard deviation of the sampling distribution for samples of size n=180. Round all answers to 3 decimal places. The population proportion is p=281,500≈0.019. This is also the mean of the sampling distribution:

μp^=p=0.019 For samples of size n=180, the standard deviation of the sampling distribution is

σpˆ=p(1−p)n−−−−−−−√=0.019(1−0.019)180−−−−−−−−−−−−−−√≈0.010

37. A dental student is conducting a study on...