Probability calculations custom essay
- The student government claims that 75%of all students favor an increase in student fees to buy indoor potted plants for the classrooms. A random sample of 9 students produced 2 in favor of the project.
What is the probability that 2 or fewer in the sample will favor the project, assuming the student government’s claim is correct? (Use 3 decimal places.)
- It is estimated that 75%of a grapefruit crop is good; the other 25% have rotten centers that cannot be detected unless the grapefruit are cut open. The grapefruit are sold in sacks of 9. Let r be the number of good grapefruit in a sack.
(c) What is the expected number of good grapefruit in a sack? (Round your answer to one decimal place)
grapefruit(d) What is the standard deviation of the r probability distribution? (Round your answer to two decimal places)
grapefruit
- A large tank of fish from a hatchery is being delivered to a lake. The hatchery claims that the mean length of fish in the tank is 15 inches, and the standard deviation is 4 A random sample of 36fish is taken from the tank. Let x be the mean sample length of these fish. What is the probability that x is within 0.5 inch of the claimed population mean? (Round your answer to four decimal places)
- Future Electronics wants to place a guarantee on the players so that no more than 10%fail during the guarantee period. Because the laser pickup is the part most likely to wear out first, the guarantee period will be based on the life of the laser beam device. How many playing hours should the guarantee cover? (Round your answer down to the nearest hour)
hr
- Assume that IQ scores are normally distributed, with a standard deviation of 17points and a mean of 100 If 70 people are chosen at random, what is the probability that the sample mean of IQ scores will not differ from the population mean by more than 2 points? (Round your answer to four decimal places)
What happens if we want several confidence intervals to hold at the same time (concurrently)? Do we still have the same level of confidence we had for each individual interval?
- Suppose we have two independent random variables x_{1}and x_{2} with respective population means μ_{1} and μ_{2}. Let us say that we use sample data to construct two 80% confidence intervals.
Confidence Interval | Confidence Level |
A_{1} < μ_{1} < B_{1} | 0.80 |
A_{2} < μ_{2} < B_{2} | 0.80 |
Now, what is the probability that both intervals hold together? Use methods of Section 5.2 to show that
P(A_{1} < μ_{1} < B_{1} and A_{2} < μ_{2} < B_{2}) = 0.64.
Hint: We are combining independent events.
P(A_{1} < μ_{1} < B_{1} and A_{2} < μ_{2} < B_{2}) = P(A_{1} < μ_{1} < B_{1}) – P(A_{2} < μ_{1} < B_{2})
P(A_{1} < μ_{1} < B_{1} and A_{2} < μ_{2} < B_{2}) = P(A_{1} < μ_{1} < B_{1}) + P(A_{2} < μ_{1} < B_{2})
P(A_{1} < μ_{1} < B_{1} and A_{2} < μ_{2} < B_{2}) = P(A_{1} < μ_{1} < B_{1}) · P(A_{2} < μ_{1} < B_{2})
P(A_{1} < μ_{1} < B_{1} and A_{2} < μ_{2} < B_{2}) = P(A_{1} < μ_{1} < B_{1}) / P(A_{2} < μ_{1} < B_{2})
If the confidence is 64% that both intervals hold together, explain why the risk that at least one interval does not hold (i.e., fails) must be 36%.
P(at least one interval fails to capture μ_{i}) = P(both intervals capture their μ_{i}) + 1
P(at least one interval fails to capture μ_{i}) = 1 – P(both intervals capture their μ_{i})
P(at least one interval fails to capture μ_{i}) = P(both intervals capture their μ_{i}) – 1
P(at least one interval fails to capture μ_{i}) = 1 + P(both intervals capture their μ_{i})
(b) Suppose we want both intervals to hold with 75% confidence (i.e., only 25% risk level). How much confidence c should each interval have to achieve this combined level of confidence? (Assume that each interval has the same confidence level c.)
Hint: | P(A_{1} < μ_{1} < B_{1} and A_{2} < μ_{2} < B_{2}) = 0.75 |
P(A_{1} < μ_{1} < B_{1} ✕ A_{2} < μ_{2} < B_{2}) = 0.75 | |
c ✕ c = 0.75 |
Now solve for c. (Use 3 decimal places.)
(c) If we want both intervals to hold at the 90% level of confidence, then the individual intervals must hold at a higher level of confidence. Write a brief but detailed explanation of how this could be of importance in a large, complex engineering design such as a rocket booster or a spacecraft.
- Three-circle, red-on-white is one distinctive pattern painted on ceramic vessels of the Anasazi period found at an archaeological site. At one excavation, a sample of 167potsherds indicated that 66were of the three-circle, red-on-white pattern.
(a) Find a point estimate p̂ for the proportion of all ceramic potsherds at this site that are of the three-circle, red-on-white pattern. (Round your answer to four decimal places)
(b) Compute a 95% confidence interval for the population proportion p of all ceramic potsherds with this distinctive pattern found at the site. (Round your answers to three decimal places)
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- A research group conducted an extensive survey of 3122wage and salaried workers on issues ranging from relationships with their bosses to household chores. The data were gathered through hour-long telephone interviews with a nationally representative sample. In response to the question, “What does success mean to you?” 1497 responded, “Personal satisfaction from doing a good job.” Let p be the population proportion of all wage and salaried workers who would respond the same way to the stated question. How large a sample is needed if we wish to be 95% confident that the sample percentage of those equating success with personal satisfaction is within 6% of the population percentage? (Hint: Use p ≈ 0.48 as a preliminary estimate. Round your answer up to the nearest whole number)
workers
- A research group conducted an extensive survey of 2955wage and salaried workers on issues ranging from relationships with their bosses to household chores. The data were gathered through hour-long telephone interviews with a nationally representative sample. In response to the question, “What does success mean to you?” 1478 responded, “Personal satisfaction from doing a good job.” Let p be the population proportion of all wage and salaried workers who would respond the same way to the stated question. Find a 90% confidence interval for p. (Round your answers to three decimal places.)
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- A machine in the student lounge dispenses coffee. The average cup of coffee is supposed to contain 7.0 ounces. A random sample ofeight cups of coffee from this machine shows the average content to be 26 ounces with a standard deviation of 0.70 ounce. Do you think that the machine has slipped out of adjustment and that the average amount of coffee per cup is different from 7 ounces? Use a 5% level of significance.
What is the value of the sample test statistic? (Round your answer to three decimal places.)(c) Find the P-value. (Round your answer to four decimal places.) |
(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α?(Answer this correctly, the dotted answer is wrong)
At the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.
At the α = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant.
At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant.
At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.
- A hospital reported that the normal death rate for patients with extensive burns (more than 40% of skin area) has been significantly reduced by the use of new fluid plasma compresses. Before the new treatment, the mortality rate for extensive burn patients was about 60%. Using the new compresses, the hospital found that only 41of 88 patients with extensive burns died. Use a 1% level of significance to test the claim that the mortality rate has dropped.
(a) What is the level of significance?
What is the value of the sample test statistic? (Round your answer to two decimal places)(c) Find the P-value. (Round your answer to four decimal places) |
(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α?
At the α = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant.
At the α = 0.01 level, we reject the null hypothesis and conclude the data are not statistically significant.
At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are statistically significant.
At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.