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1 Identify each of the random variables described below as d

1. Identify each of the random variables described below as discrete or continuous
a. The number of fountain drinks purchased by patrons at a local quick store.
b. The time it takes for a randomly chosen MATH 250 student to create an EXCEL formula to determine the Average of a column of numbers.
c. The number of automobiles owned by a section of MATH 250 students.
d. The arm span of a randomly selected MATH 250 student.
2. The probability distribution table at the right is of the discrete random variable x representing the number of cars owned by each family in a specific neighborhood of Wichita, KS. Answer the following:
x P(X=x)
(a) Determine the value that is missing in the table. (Hint: what are the requirments for a probability distribution?) 0 0.12
1
2 0.24
(b) Find the probability that x is at least 2 , that is find P(x 2). 3 0.11
4 0.07
(c) Find P(x < 1). Describe what the resulting value represents within the given context.
(d) Find the mean (expected value) and standard deviation of this probability distribution.
3. (a) What is meant by the term \"expected value\"?
(b) Suppose the expected value of a game of chance is -$2.75. Does this mean that every time one plays this game, she will lose $2.75? Explain your answer briefly.
4. List the four requirements needed for an experiment/procedure to be considered a binomial distribution?
5. A company produces a device for the purposes of medical research. As with all production lines of senstive equipment, even when production processes are working correctly, not all devices sent out from the factory are flawless. Suppose the company has found that there is a 15% rate of defect on these devices after shipment. You currently received a shipment of 14 such devices from this company. Use this situation to answer the parts below.
(a) Recognizing that this is a binomial situation, give the meaning S and F in this context. That is, define in words what you will classify as a successful trial and what you will classify as a failure trial when a device is selected and tested for defects.
S =
F =
(a) Next, give the values of n, p, and q.
(b) Construct the complete binomial probability distribution for this situation in a table out to the right.
(c) Using your table, find the probability that exactly two of the randomly selected devices is defective.
(d) Find the probability that at least three of the devices are defective.
(e) Find the probability that less than two of the devices are defective.
(f) Find the mean and standard deviation of this binomial probability distribution.  
(g) By writing a sentence, interpret the meaning of the mean value found in (f) as tied to the context of defective devices in a shipment of 14.
(h) Is it unusual to have all 14 of the devices in a shipment work correctly? Briefly explain your answer giving supporting numerical evidence.
6. What is a normal distribution? What is a standard normal distribution? Briefly compare and contrast these two statistical terms.
7. With regards to a standard normal distribution complete the following:
(a) Find P(z < -1.75), the percentage of the standard normal distribution below the z-score of -1.75.
(b) Find P(z < 2.2), the percentage of the standard normal distribution below the z-score of 2.2.
(c) Find P(-1.85 < z < 1.96).
(d) Find P( z > 2.0).
(e) Find the z-score that separates the lower 75% of standarized scores from the top 25% . . . that is find the z-score corresponding to P75, the 75th percentile value in the standard normal distribution.
8. If the results on a nationally administered university entry exam are normally distributed with a mean of 45 points and a standard deviation of 4 points, determine the following:
(a) Describe the graph of this distribution (if you can do so, produce an electronic sketch of the graph to the right, otherwise adequately describe the distribution graph through its shape and horizontal scale values.)
(b) Find the z-score for a single exam that had 38 points. Then find the z-score for one with 45 points.
(c) If x represents a possible point-score on the exam, find P(x < 38).
(d) Find P(38 < x < 52) and give an interpretation of this value.
(e) Suppose a certain university requires one to score in the top 35% of all such scores to be admitted. What is the minimum number of points one must score on this exam for admission?
9. Explain what a “sampling distribution of sample means” is. Be specific!
10. Fill in the blanks in the statements below.
The Central Limit Theorem states that as the sample size increases, the distribution of all the possible sample means (that is the sampling distribution of sample means) can be approximated by a ________________ distribution. The mean in the sampling distribution will be the same as the ________________ mean and the standard deviation in the sampling distribution will be ________________ .
1. Identify each of the random variables described below as discrete or continuous
a. The number of fountain drinks purchased by patrons at a local quick store.
b. The time it takes for a randomly chosen MATH 250 student to create an EXCEL formula to determine the Average of a column of numbers.
c. The number of automobiles owned by a section of MATH 250 students.
d. The arm span of a randomly selected MATH 250 student.
2. The probability distribution table at the right is of the discrete random variable x representing the number of cars owned by each family in a specific neighborhood of Wichita, KS. Answer the following:
x P(X=x)
(a) Determine the value that is missing in the table. (Hint: what are the requirments for a probability distribution?) 0 0.12
1
2 0.24
(b) Find the probability that x is at least 2 , that is find P(x 2). 3 0.11
4 0.07
(c) Find P(x < 1). Describe what the resulting value represents within the given context.
(d) Find the mean (expected value) and standard deviation of this probability distribution.
3. (a) What is meant by the term \"expected value\"?
(b) Suppose the expected value of a game of chance is -$2.75. Does this mean that every time one plays this game, she will lose $2.75? Explain your answer briefly.
4. List the four requirements needed for an experiment/procedure to be considered a binomial distribution?
5. A company produces a device for the purposes of medical research. As with all production lines of senstive equipment, even when production processes are working correctly, not all devices sent out from the factory are flawless. Suppose the company has found that there is a 15% rate of defect on these devices after shipment. You currently received a shipment of 14 such devices from this company. Use this situation to answer the parts below.
(a) Recognizing that this is a binomial situation, give the meaning S and F in this context. That is, define in words what you will classify as a successful trial and what you will classify as a failure trial when a device is selected and tested for defects.
S =
F =
(a) Next, give the values of n, p, and q.
(b) Construct the complete binomial probability distribution for this situation in a table out to the right.
(c) Using your table, find the probability that exactly two of the randomly selected devices is defective.
(d) Find the probability that at least three of the devices are defective.
(e) Find the probability that less than two of the devices are defective.
(f) Find the mean and standard deviation of this binomial probability distribution.  
(g) By writing a sentence, interpret the meaning of the mean value found in (f) as tied to the context of defective devices in a shipment of 14.
(h) Is it unusual to have all 14 of the devices in a shipment work correctly? Briefly explain your answer giving supporting numerical evidence.
6. What is a normal distribution? What is a standard normal distribution? Briefly compare and contrast these two statistical terms.
7. With regards to a standard normal distribution complete the following:
(a) Find P(z < -1.75), the percentage of the standard normal distribution below the z-score of -1.75.
(b) Find P(z < 2.2), the percentage of the standard normal distribution below the z-score of 2.2.
(c) Find P(-1.85 < z < 1.96).
(d) Find P( z > 2.0).
(e) Find the z-score that separates the lower 75% of standarized scores from the top 25% . . . that is find the z-score corresponding to P75, the 75th percentile value in the standard normal distribution.
8. If the results on a nationally administered university entry exam are normally distributed with a mean of 45 points and a standard deviation of 4 points, determine the following:
(a) Describe the graph of this distribution (if you can do so, produce an electronic sketch of the graph to the right, otherwise adequately describe the distribution graph through its shape and horizontal scale values.)
(b) Find the z-score for a single exam that had 38 points. Then find the z-score for one with 45 points.
(c) If x represents a possible point-score on the exam, find P(x < 38).
(d) Find P(38 < x < 52) and give an interpretation of this value.
(e) Suppose a certain university requires one to score in the top 35% of all such scores to be admitted. What is the minimum number of points one must score on this exam for admission?
9. Explain what a “sampling distribution of sample means” is. Be specific!
10. Fill in the blanks in the statements below.
The Central Limit Theorem states that as the sample size increases, the distribution of all the possible sample means (that is the sampling distribution of sample means) can be approximated by a ________________ distribution. The mean in the sampling distribution will be the same as the ________________ mean and the standard deviation in the sampling distribution will be ________________ .

Solution

1) a) The number of fountain drinks purchased by patrons is discrete

1. Identify each of the random variables described below as discrete or continuous a. The number of fountain drinks purchased by patrons at a local quick store
1. Identify each of the random variables described below as discrete or continuous a. The number of fountain drinks purchased by patrons at a local quick store
1. Identify each of the random variables described below as discrete or continuous a. The number of fountain drinks purchased by patrons at a local quick store
1. Identify each of the random variables described below as discrete or continuous a. The number of fountain drinks purchased by patrons at a local quick store
1. Identify each of the random variables described below as discrete or continuous a. The number of fountain drinks purchased by patrons at a local quick store
1. Identify each of the random variables described below as discrete or continuous a. The number of fountain drinks purchased by patrons at a local quick store

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