The reading speed of second grade students in a large city is approximately normal, with a mean of
92 words per minute (wpm) and a standard deviation of 10 wpm.
(a) What is the probability a randomly selected student in the city will read more than 97 words per minute?
Interpret this probability.
(b) What is the probability that a random sample of 12 second grade students from the city results in a mean reading rate of more than 97 words per minute?
Interpret this probability.
(c) What is the probability that a random sample of 24 second grade students from the city results in a mean reading rate of more than 97 words per minute?
Interpret this probability.
(d) What effect does increasing the sample size have on the probability? Provide an explanation for this result.
(e) A teacher instituted a new reading program at school. After 10 weeks in the program, it was found that the mean reading speed of a random sample of 21 second grade students was 94.3 wpm. What might you conclude based on this result?
(f) There is a 5% chance that the mean reading speed of a random sample of 25 second grade students will exceed what value?
In: Statistics and Probability
A subway has good service 70% of the time and runs less frequently 30% of the time because of signal problems. When there are signal problems, the amount of time in minutes that you have to wait at the platform is described by the pdf probability density function with signal problems = pT|SP(t) = .1e −.1t But when there is good service, the amount of time you have to wait at the platform is probability density function with good service = pT|Good(t) = .3e −.3t You arrive at the subway platform and you do not know if the train has signal problems or is running with good service, so there is a 30% chance the train is having signal problems. (a) What is the probability that you wait at least 1 minute if there is good service? (b) What is the probability that you wait at least 1 minute if there are signal problems? (c) After 1 minute of waiting on the platform, you decide to re-calculate the probability that there are signal problems conditioning on the fact that your wait will be at least 1 minute long (since you have already waited 1 minute). What is that new probability? (d) After 5 minutes of waiting, still no train. You re-calculate again. What is the new probability?
In: Statistics and Probability
The research team in your marketing department has some preliminary results for the upcoming launch of your new product. They believe that there is a 90% chance that the launch of your new product will be successful if the state of the economy is in good health. Conversely, they believe that there is only a 30% chance that the launch will be successful if the economy is not in good health. Past indicators suggest that the probability that the economy will be in good health (during the launch period) is 80%.
1) What is the probability that the product launch will be
successful and the economy will be in good health?
a) 0.90 b) 0.72 c) 0.80 d) 0.92
2) What is the probability that the product launch will not be
successful and the economy will not be in good health?
a) 0.14 b) 0.63 c) 0.70 d) 0.20
3) What is the probability that the product launch will be successful? a) 0.78 b) 0.90 c) 0.80 d) 0.86
4) Given that the launch was successful, what is the probability
that the economy was in good health?
a) 0.90 b) 0.72 c) 0.80 d) 0.92
5) Given that the launch was not successful, what is the
probability that the economy was in good health?
a) 0.20 b) 0.30 c) 0.36 d) 0.08
6) Given that the economy is in good health, what is the
probability that the launch of your product will not be
successful?
a) 0.10 b) 0.30 c) 0.20 d) 0.36
In: Statistics and Probability
In: Statistics and Probability
1. A pro basketball player is a poor free-throw shooter. Consider situations in which he shoots a pair of free throws. The probability that he makes the first free throw is 0.51. Given that he makes the first, suppose the probability that he makes the second is 0.64. Given that he misses the first, suppose the probability that he makes the second one is 0.36.
What is the Probability he makes both free throws?
2. The most likely scenario for an accident for a natural gas pipeline is natural-gas leakage from a hole in the pipeline. The probability that the leak ignites immediately (I) causing a jet fire is .03. If the leak does not immediately ignite, it may result in the delayed ignition (D) of a gas cloud. If there is no delayed ignition, the gas cloud will disperse harmlessly (H). Given no immediate ignition, the probability of delayed ignition causing a flash fire is .03. Suppose a leak occurs in the natural-gas pipeline.
The probability that either a jet fire or a flash fire will occur is?
3. Red snapper is a rare and expensive reef fish served at upscale restaurants. A certain law prohibits restaurants from serving a cheaper, look-alike variety of fish (vermilion snapper or lane snapper) to customers who order red snapper. Researchers at a university used DNA analysis to examine fish specimens labeled "red snapper" that were purchased form vendors across the country. The DNA tests revealed that 78% of the specimens were not red snapper, but the cheaper, look-alike variety of fish.
Assuming that the results of the DNA analysis are valid, what is the probability that you are actually served red snapper the next time you order it at a restaurant?
The probability is?
In: Statistics and Probability
According to a survey, the probability that a randomly selected worker primarily drives a car to work is 0.896. The probability that a randomly selected worker primarily takes public transportation to work is 0.033. Complete parts (a) through (d).
(a) What is the probability that a randomly selected worker primarily drives a car or takes public transportation to work? P(worker drives a car or takes public transportation to work)equals nothing (Type an integer or decimal rounded to three decimal places as needed.)
(b) What is the probability that a randomly selected worker primarily neither drives a car nor takes public transportation to work? P(worker neither drives a car nor takes public transportation to work)equals nothing (Type an integer or decimal rounded to three decimal places as needed.)
(c) What is the probability that a randomly selected worker primarily does not drive a car to work? P(worker does not drive a car to work)equals nothing (Type an integer or decimal rounded to three decimal places as needed.)
(d) Can the probability that a randomly selected worker primarily walks to work equal 0.15? Why or why not? A. No. The probability a worker primarily drives, walks, or takes public transportation would be less than 1. B. Yes. The probability a worker primarily drives, walks, or takes public transportation would equal 1. C. No. The probability a worker primarily drives, walks, or takes public transportation would be greater than 1. D. Yes. If a worker did not primarily drive or take public transportation, the only other method to arrive at work would be to walk.
In: Statistics and Probability
The judges of County X try thousands of cases per year. Although in a big majority of the cases disposed the verdict stands as rendered, some cases are appealed. Of those appealed, some are reversed. Because appeals are often made as a result of mistakes by the judges, you want to determine which judges are doing a good job and which ones are making too many mistakes. The attached Excel file has the results of 182,908 disposed cases over a three year period by the 38 judges in various courts of County X. Two of the judges (Judge 3 and Judge 4) did not serve in the same court for the entire three-year period. Using your knowledge of probability and conditional probability you will make an analysis to decide a ranking of judges. You will also analyze the likelihood of appeal and reversal for cases handled by different courts. Using Excel to calculate the following probabilities for each judge. Use the attached Excel file to fill in these numbers on the tables for each court. Each court is given on a separate tab. The probability of a case being appealed for each judge. The probability of a case being reversed for each judge. The probability of a reversal given an appeal for each judge. Probability of cases being appealed and reversed in the three courts. Rank the judges within each court.
| Domestic Relations Court | |||||||
| Judge | Total Cases Disposed | Appealed Cases | Reversed Cases | Probability of Appeal | Probability of Reversal | Conditional Probability of Reversal Given Appeal | Rank of Judge |
| Judge 17 | 2,729 | 7 | 1 | ||||
| Judge 3 | 6,001 | 19 | 4 | ||||
| Judge 18 | 8,799 | 48 | 9 | ||||
| Judge 19 | 13,970 | 32 | 3 | ||||
| Total | 31,499 | 106 | 17 | ||||
In: Math
Suppose a geyser has a mean time between eruptions of 65 minutes . Let the interval of time between the eruptions be normally distributed with standard deviation 26 minutes . Complete parts ?(a) through ?(e) below.
(a) What is the probability that a randomly selected time interval between eruptions is longer than
76 ?minutes?
The probability that a randomly selected time interval is longer than
76
minutes is approximately
nothing
.
?(Round to four decimal places as? needed.)
?(b) What is the probability that a random sample of
16
time intervals between eruptions has a mean longer than
76
?minutes?
The probability that the mean of a random sample of
16
time intervals is more than
76
minutes is approximately
nothing
.
?(Round to four decimal places as? needed.)
?(c) What is the probability that a random sample of
31
time intervals between eruptions has a mean longer than
76
?minutes?
The probability that the mean of a random sample of
31
time intervals is more than
76
minutes is approximately
nothing
.
?(Round to four decimal places as? needed.)
?(d) What effect does increasing the sample size have on the? probability? Provide an explanation for this result. Fill in the blanks below.
If the population mean is less than
76
?minutes, then the probability that the sample mean of the time between eruptions is greater than
76
minutes
?
increases
decreases
because the variability in the sample mean
?
increases
decreases
as the sample size
?
decreases.
increases.
?(e) What might you conclude if a random sample of
31
time intervals between eruptions has a mean longer than
76
?minutes? Select all that apply.
A.
The population mean must be more than
65
?,
since the probability is so low.
B.
The population mean is
65
?,
and this is just a rare sampling.
C.
The population mean is
65
?,
and this is an example of a typical sampling result.
D.
The population mean may be less than
65
.
E.
The population mean may be greater than
65
.
F.
The population mean cannot be
65
?,
since the probability is so low.
G.
The population mean must be less than
65
?,
since the probability is so low.
In: Statistics and Probability
A. Suppose x has a distribution with μ = 23 and σ = 15.
(a) If a random sample of size n = 39 is drawn, find μx, σx and P(23 ≤ x ≤ 25). (Round σx to two decimal places and the probability to four decimal places.)
| μx = |
| σx = |
| P(23 ≤ x ≤ 25) = |
(b) If a random sample of size n = 64 is drawn, find
μx, σx
and P(23 ≤ x ≤ 25). (Round
σx to two decimal places and the
probability to four decimal places.)
| μx = |
| σx = |
| P(23 ≤ x ≤ 25) = |
(c) Why should you expect the probability of part (b) to be higher
than that of part (a)? (Hint: Consider the standard
deviations in parts (a) and (b).)
The standard deviation of part (b) is ---Select--- the
same as larger than smaller than part (a) because of
the ---Select--- larger smaller same sample size.
Therefore, the distribution about μx
is ---Select--- wider the same narrower
B. Suppose the heights of 18-year-old men are approximately normally distributed, with mean 67 inches and standard deviation 2 inches.
(a) What is the probability that an 18-year-old man selected at
random is between 66 and 68 inches tall? (Round your answer to four
decimal places.)
(b) If a random sample of twenty-nine 18-year-old men is selected,
what is the probability that the mean height x is between
66 and 68 inches? (Round your answer to four decimal places.)
(c) Compare your answers to parts (a) and (b). Is the probability
in part (b) much higher? Why would you expect this?
The probability in part (b) is much higher because the mean is larger for the x distribution.The probability in part (b) is much higher because the standard deviation is smaller for the x distribution. The probability in part (b) is much lower because the standard deviation is smaller for the x distribution.The probability in part (b) is much higher because the mean is smaller for the x distribution.The probability in part (b) is much higher because the standard deviation is larger for the x distribution.
In: Statistics and Probability
Suppose a geyser has a mean time between eruptions of 71 minutes.Let the interval of time between the eruptions be normally distributed with standard deviation 24 minutes.
Complete parts (a) through (e) below.
(a) What is the probability that a randomly selected time interval between eruptions is longer than 83 minutes?
The probability that a randomly selected time interval is longer than 83 minutes is approximately ____
(Round to four decimal places as needed.)
(b) What is the probability that a random sample of 8 time intervals between eruptions has a mean longer than 83 minutes?
The probability that the mean of a random sample of 8 time intervals is more than 83 minutes is approximately ______
(Round to four decimal places as needed.)
(c) What is the probability that a random sample of 24 time intervals between eruptions has a mean longer than 83 minutes?
The probability that the mean of a random sample of 24 time intervals is more than 83 minutes is approximately ____
(Round to four decimal places as needed.)
(d) What effect does increasing the sample size have on the probability? Provide an explanation for this result. Fill in the blanks below.
If the population mean is less than 83 minutes, then the probability that the sample mean of the time between eruptions is greater than
83 minutes (________________)because the variability in the sample mean(___________)as the sample size (_____________)
(e) What might you conclude if a random sample of 24 time intervals between eruptions has a mean longer than 83 minutes? Select all that apply
A.The population mean is71,and this is an example of a typical sampling result.
B.The population mean is 71,and this is just a rare sampling.
C.The population mean may be greater than 71
D.The population mean must be more than 71,since the probability is so low.
E.The population mean cannot be 71,since the probability is so low.
F.The population mean must be less than 71 since the probability is so low.
G.The population mean may be less than 71
In: Statistics and Probability