1. Identify each of the random variables described below as discrete or continuous
a. The distance a baseball travels in the air after being hit.
b. The number televisions sold in the next week at the local electronics store.
c. The square footage of your home/apartment.
d. The number of people in class with blood type A.
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:
(a) Determine the value that is missing in the table. (Hint: what are the requirements for a probability distribution?)
(b) Find the probability that x is at least 2 , that is find P(x ≥ 2).
(c) Find P(x =0). Describe what the resulting value represents within the given context.
(d) Find the mean μ (expected value) and standard deviation σ 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 sensitive 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 5% rate of defect on these devices after shipment. You currently received a shipment of 12 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.
(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 twelve randomly selected devices is defective.
(d) Find the probability that at least three of the twelve 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 12.
(h) Is it unusual to have all 12 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 < -1.5), the percentage of the standard normal distribution below the z-score of -1.5.
(b) Find P(z < 2.0), the percentage of the standard normal distribution below the z-score of 2.
(c) Find P(-1.25 < z < 1.75).
(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 ________________ .
11. Explain the difference between the two following symbols:
12. An medical research firm has designed a machine which returns a number as it tests for abnormal blood antibodies. Tests by this machine of normal "healthy" blood produces numerical values that are normally distributed with a mean of 50 and a standard deviation of 6.5.
(a) Describe the shape and horizontal scaling on the graph of the distribution for the population of all tests. (Hint--parts (a) and (b) here are related back to the ideas in section 6-3 of the text.)
(b) Find the probability that the score on the blood test from a randomly selected individual will be less than 40---find P(x < 40). Based upon your result, discuss if it seems likely to randomly select an individual whose blood test score is less than 40?
(c) Suppose all possible samples of size fifty from the population of all test results are drawn and the mean is found for each resulting sample. Describe the shape and scaling on the graph of the resulting sampling distribution for the sample means. (Hint: Apply the Central Limit Theorem!)
(d) Find the probability that the mean on the blood test from a randomly produced sample of fifty scores will be less than 40---find P(sample mean < 40). Based upon your result, discuss if it seems likely for a collection of fifty randomly selected scores to be produce a mean value that is less than 40?
(e) Find the probability that the mean from a sample of fifty scores on this blood test will be between 45 and 55. Based upon your result, does it seem likely to have a collection of fifty scores produce a mean within this interval of 45 to 55?
13. Suppose that a 95% confidence interval for a population mean μ is given to be from 130 to 150. The 95% value in this statement indicates that: (choose the one best answer)
14. Suppose that a 90% confidence interval for a population mean μ is constructed from a sample, resulting in the interval from 120 to 160. What was the value of the sample mean that was collected? What was the size of the margin of error?
15. What is the appropriate critical z-value for each of the following confidence levels?
(a) 98% confidence level:
(b) 90% confidence level:
16. What happens to the size (width) of a confidence interval in a problem if:
(a) the percent of desired confidence decreases from 95% to 90%? Does it get wider or narrower?
17. A survey of 600 randomly selected adults across the US revealed that 450 supported the building of electric generating windmills. You want to produce a 96% confidence interval for the percentage of ALL adults from US with this same viewpoint.
(a) Determine the Margin of Error for this situation.
(b) Determine the confidence interval for this situation.
(c) Give an interpretive sentence to your results in part (b).
18. Interested in social experiments, you want to estimate the percentage of US adults willing to sing karaoke in front of strangers. You want to be within 2 percentage points of the true proportion measure, based upon a 95% confidence level. What size sample is required?--(NOTE that no estimate of "p-hat" is given.)
19. A farmer needs to determine the mean weight of all eggs produced by her chickens after she changed their diets. A random sample of 250 eggs yields a mean of 1.9 ounces. Based upon a desired 95% confidence level, determine the associated confidence interval. (Assume a population standard deviation of 0.45 oz.)
20. You need to estimate the mean Christmas sales across a large chain of jewelry stores. You want to estimate this mean value to within $50 with 99% confidence. Previous studies indicate that the standard deviation for sales across the stores has been $200. How many randomly selected stores from the chain should your sample contain to meet this desired level of accuracy?
Attachment:- Assignment.rar