Question 1

Define a density curve and describe the two properties that it must satisfy. Show a density curve for a uniform distribution. Make sure that your graph satisfies both properties.

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Question 2

Three randomly selected households are surveyed as a pilot project for a larger survey to be conducted later. The numbers of people in the households are 1, 4, and 8. Consider the values of 1, 4, and 8 to be a population. Assume that samples of size n = 2 are randomly selected with replacement from the population of 1, 4, and 8. The nine different samples are as follows: (1, 1), (1, 4), (1, 8), (4, 1), (4, 4), (4, 8), (8, 1), (8, 4), and (8, 8).
(i) Find the median of each of the nine samples, then summarize the sampling distribution of the medians in the format of a table representing the probability distribution. 
(ii) Compare the population median to the mean of the sample medians. 
(iii) Do the sample medians target the value of the population median? In general, do medians make good estimators of population medians? Why or why not?

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Question 3

In a certain population, 10% of people are left handed. Suppose that in calculating each of the probabilities below, you use the normal distribution as an approximation to the binomial but that you fail to use a continuity correction. In which case will the resulting error be the greatest? In which case will the error be the least? Explain your thinking.
A: the probability that among 50 randomly selected people, at least 5 are left handed
B: the probability that among 100 randomly selected people, more than 10 are left handed
C: the probability that among 200 randomly selected people, at most 20 are left handed

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Question 4

The ages (in years) of the four U.S. vice presidents who assumed office after presidential assassinations are 56 (A. Johnson), 51 (C. Arthur), 42 (T. Roosevelt), and 55 (L.B. Johnson).
(i) Assuming that 2 of the ages are randomly selected with replacement, list the 16 different possible samples. 
(ii) Find the range of each of the 16 samples, then summarize the sampling distribution of the ranges in the format of a table representing the probability distribution. 
(iii) Compare the population range to the mean of the sample ranges.
 (iv) Do the sample ranges target the value of the population range? In general, do sample ranges make good estimators of population ranges? Why or why not?

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Question 5

A normal quartile plot is given below for the weekly incomes (in dollars) of a sample of engineers in one town. Use the plot to assess the normality of the incomes of engineers in this town. Explain your reasoning.

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Question 6

Examine the given data set and determine whether the requirement of a normal distribution is satisfied. Assume that the requirement for a normal distribution is loose in the sense that the population distribution need not be exactly normal, but it must have a distribution which is basically symmetric with only one mode. Explain why you do or do not think that the requirement is satisfied.
-The heart rates (in beats per minute) of 30 randomly selected students are given below. $$\begin{array} { l l l l l } 
78 & 64 & 69 & 75 & 80 \
63 & 70 & 72 & 72 & 68 \
77 & 71 & 74 & 84 & 70 \
62 & 67 & 71 & 69 & 58 \
74 & 70 & 80 & 63 & 88 \
60 & 68 & 69 & 70 & 71
\end{array}$$

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Question 7

Three randomly selected households are surveyed as a pilot project for a larger survey to be conducted later. The numbers of people in the households are 1, 3, and 9. Consider the values of 1, 3, and 9 to be a population. Assume that samples of size n = 2 are randomly selected with replacement from the population of 1, 3, and 9. The nine different samples are as follows: (1, 1), (1, 3), (1, 9), (3, 1), (3, 3), (3, 9), (9, 1), (9, 3), and (9, 9). 
(i) Find the variance of each of the nine samples, then summarize the sampling distribution of the variances in the format of a table representing the probability distribution. 
(ii) Compare the population variance to the mean of the sample variances. 
(iii) Do the sample variances target the value of the population variance? In general, do variances make good estimators of population variances? Why or why not?

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Question 8

A normal quartile plot is given below for the lifetimes (in hours) of a sample of batteries of a particular brand. Use the plot to assess the normality of the lifetimes of these batteries. Explain your reasoning.

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Question 9

Suppose you are asked to find the 20th percentile and the 80th percentile for a set of scores.
These two problems are solved almost exactly the same. Draw the diagram for each and discuss the part of the solution that would be different in finding the requested probabilities.

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Question 10

Construct a normal quartile plot of the given data. Use your plot to determine whether the data come from a normally distributed population.
-The weekly incomes (in dollars) of a sample of 12 nurses working at a Los Angeles hospital
are given below.

$\begin{array}{rrrr}
500 & 750 & 630 & 480 \
550 & 650 & 720 & 780 \
820 & 960 & 1200 & 770
\end{array}$

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Question 11

According to data from the American Medical Association, 10% of us are left-handed.
Suppose groups of 500 people are randomly selected. Find the probability that at least 80 are left-handed. Describe the characteristics of this problem which help you to recognize that the problem is about a binomial distribution which you are to solve by estimating with the normal distribution. (Assume that you would not use a computer, a table, or the binomial probability formula.)

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Question 12

Explain how a nonstandard normal distribution differs from the standard normal distribution. Describe the process for finding probabilities for nonstandard normal distributions.

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Question 13

SAT verbal scores are normally distributed with a mean of 430 and a standard deviation of 120 (based on data from the College Board ATP). (a) If a single student is randomly selected, find the probability that the sample mean is above 500. (b) If a sample of 35 students are selected randomly, find the probability that the sample mean is above 500.
These two problems appear to be very similar. Which problem requires the application of the central limit theorem, and in what way does the solution process differ between the two problems?

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Question 14

The number of books sold over the course of the four-day book fair were 134, 178, 268, and 58. Assume that samples of size 2 are randomly selected with replacement from this population of four values. List the different possible samples, and find the mean of each of them.

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Question 15

The ages (in years) of the four U.S. vice presidents who assumed office after presidential assassinations are 56 (A. Johnson), 51 (C. Arthur), 42 (T. Roosevelt), and 55 (L.B. Johnson).
(i) Assuming that 2 of the ages are randomly selected with replacement, list the 16 different possible samples. 
(ii) Find the standard deviation of each of the 16 samples, then summarize the sampling distribution of the standard deviation in the format of a table representing the probability distribution. 
(iii) Compare the population standard deviation to the mean of the sample standard deviations. 
(iv) Do the sample standard deviations target the value of the population standard deviation? In general, do sample standard deviations make good estimators of population standard deviations? Why or why not?

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Question 16

A poll of 1900 randomly selected students in grades 6 through 8 was conducted and found that 37% enjoy playing sports. Would confidence in the results increase if the sample size were 3500 instead of 1900? Why or why not?

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Question 17

Explain why a continuity correction factor is necessary when approximating the binomial distribution by the normal distribution. Refer to the terms "discrete" and continuous", and draw a diagram to support your answer.

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Question 18

The ages (in years) of the four U.S. vice presidents who assumed office after presidential assassinations are 56 (A. Johnson), 51 (C. Arthur), 42 (T. Roosevelt), and 55 (L.B. Johnson).
(i) Assuming that 2 of the ages are randomly selected with replacement, list the 16 different possible samples. 
(ii) Find the variance of each of the 16 samples, then summarize the sampling distribution of the variances in the format of a table representing the probability distribution. 
(iii) Compare the population variance to the mean of the sample variances.
(iv) Do the sample variances target the value of the population variance? In general, do sample variances make good estimators of population variances? Why or why not?

Accepted Answer

The answer of The ages (in years) of the four...

Question 19

Personal phone calls received in the last three days by a new employee were 2, 6, and 3. Assume that samples of size 2 are randomly selected with replacement from this population of three values. Identify the probability of each sample, and describe the sampling distribution of the sample means.

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Question 20

Examine the given data set and determine whether the requirement of a normal distribution is satisfied. Assume that the requirement for a normal distribution is loose in the sense that the population distribution need not be exactly normal, but it must have a distribution which is basically symmetric with only one mode. Explain why you do or do not think that the requirement is satisfied.
-The data below represents the amount of television watched per week (in hours) for 40 randomly selected teenagers. 
$$\begin{array} { r r r r r r r r r r } 
13 & 4 & 17 & 14 & 9 & 6 & 7 & 5 & 14 & 12 \
20 & 16 & 0 & 15 & 10 & 6 & 5 & 3 & 13 & 14 \
15 & 5 & 3 & 5 & 8 & 11 & 12 & 13 & 14 & 7 \
4 & 6 & 9 & 13 & 3 & 14 & 24 & 15 & 17 & 20
\end{array}$$

Accepted Answer

The answer of Examine the given data set and determine...

Quiz 6: Normal Probability Distributions

Define a density curve and describe the two properties that it must satisfy. Show a density curve for a uniform distribution. Make sure that your graph satisfies both properties.

A normal quartile plot is given below for the weekly incomes (in dollars) of a sample of engineers in one town. Use the plot to assess the normality of the incomes of engineers in this town. Explain your reasoning.

A normal quartile plot is given below for the lifetimes (in hours) of a sample of batteries of a particular brand. Use the plot to assess the normality of the lifetimes of these batteries. Explain your reasoning.

Suppose you are asked to find the 20th percentile and the 80th percentile for a set of scores. These two problems are solved almost exactly the same. Draw the diagram for each and discuss the part of the solution that would be different in finding the requested probabilities.

Explain how a nonstandard normal distribution differs from the standard normal distribution. Describe the process for finding probabilities for nonstandard normal distributions.

The number of books sold over the course of the four-day book fair were 134, 178, 268, and 58. Assume that samples of size 2 are randomly selected with replacement from this population of four values. List the different possible samples, and find the mean of each of them.

A poll of 1900 randomly selected students in grades 6 through 8 was conducted and found that 37% enjoy playing sports. Would confidence in the results increase if the sample size were 3500 instead of 1900? Why or why not?

Explain why a continuity correction factor is necessary when approximating the binomial distribution by the normal distribution. Refer to the terms "discrete" and continuous", and draw a diagram to support your answer.

Personal phone calls received in the last three days by a new employee were 2, 6, and 3. Assume that samples of size 2 are randomly selected with replacement from this population of three values. Identify the probability of each sample, and describe the sampling distribution of the sample means.