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# Probability of Outcomes of Different Situations

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Five percent of patients suffering from a certain disease are selected to undergo a new treatment that is believed to increase the recovery rate from 30% to 50%. A person is randomly selected from these patients after the completion of the treatment and is found to have recovered. What is the probability that the patient received the new treatment?

An executive has three telephones on her desk, each with a different number, and none is an extension. During a 15-second interval the probability that phone 1 rings is .35, that phone 2 rings is .60, and that phone 3 rings is .10. find the probability that during this interval of time
All three phones ring.
At least one phone rings.
At most two phones ring.
None of the phones ring.
Let ƒ be a symmetric density. Let a > 0. Show that:
F(0)=1/2, where F is the distribution function.
P(X>a)=1/2- where a=0, b=a.
F(-a) + F(a) =1.
P(lXl > a) = 2F(-a) = 2[1-F(a)].
P(lXl < a) = 2F(a) -1.
Which of the following functions are joint density functions?
f(x,y)=e-(x+y), x>0, y>0, and zero elsewhere.
f(x,y)=8x2y, 0<y<x<1, and zero elsewhere.
A point X is chosen from the interval [0,1] according to the density function ƒ1(given below). Another point Y is chosen from the interval [X,1] (so that X< Y< 1) according to the density function g(ylx). Let
ƒ1(x) =1, 0<x<1, and zero elsewhere, and for 0< x < 1
g(ylx)= 1/(1-x) if x< y <1, and zero elsewhere.
Find the joint density of X and Y and the marginal density of Y.
Find the conditional density of X given Y=y.
A fair die is rolled twice. Let X be the sum of face values and Y the absolute value of the difference in face values on the two rolls. Are X and Y independent? Please justify the answer.
The acidity of a compound X depends on the proportion Y of a certain chemical present in the compound. Suppose
X = (α + βY)2
Where α, β are constants. If the density of Y can be assumed to be
g( y) = 2y, 0< y < 1, and zero elsewhere
Find the average acidity of the compound and Var(X).
For each of the following distributions, find a (the) median.
f (x) = 1/ (σ√2π)exp{-(x- μ)2/ 2 σ2}; -∞ < x <∞, μ ∈R
f(x)= 1/x2, x > 1, and zero elsewhere.

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## Hypothesis

1
Identify the null hypothesis, alternative hypothesis, test statistic, P-value or critical value(s), conclusion about the null hypothesis, and final conclusion that addresses the original claim.

a) The Genetics and IVF Institute conducted a clinical trial of the YSORT method designed to increase the probability of conceiving a boy. As this book was being written, 51 babies were born to parents using the YSORT method, and 39 of them were boys. Use the sample data with a 0.01 significance level to test the claim that with this method, the probability of a baby being a boy is greater than 0.5. Does the method appear to work?

2
A simple random sample consists of n=12 values, so the requirement of normality must be checked. How can you check that requirement of normality?

3
When consumers apply for credit, their credit is rated using FICO scores. A random sample of credit ratings is obtained, and the FICO scores are summarized with these statistics: n=18, x=2700g, s = 645g. Use a 0.05 significance level to test the claim that these credit ratings are from a population with a mean that is equal to 700. If the Bank of Newport requires a credit ratting of 700 or higher for a car loan, do the results indicate that everyone will be eligible for a car loan? Why?

4
The Stewart Aviation Products Company uses a new production method to manufacture aircraft altimeters. A simple random sample of 81 altimeters is tested in a pressure chamber, and the errors in altitude are recorded as positive values or negative values. The sample has a standard deviation of s=52.3 ft. At the 0.05 significance level, test the claim that the new production line has errors with a standard deviation different from 43.7ft which was the standard deviation for the old production method appear to be better or worse than the old method?

5
When respondents are asked a question on a survey, 40 of them answer yes, 60 of them answer no, and there are no other responses. What is the sample proportion of yes responses, and what notation is used to represent it?

6
In a survey of 703 randomly selected workers, 15.93% got their jobs through newspaper ads. Consider a hypothesis test that uses a 0.05 significance level to test the claim that less than 20% of workers get their jobs through newspaper ads.
What is the test statistic?
What is the critical value?
What is the P-value?
What is the conclusion?
Based on the preceding results, can we conclude that 15.93% is significantly less than 20% for all such hypothesis tests? Why?

7
Determine the given conditions justify using the methods of this section when testing a claim about the population mean µ?

a) The sample size is n=9, &#963; = 2.5, and a histogram of the sample data shows a distribution that is far from being bell-shaped.

b) The sample size is n=121, &#963;= 0.25, and a histogram of the sample data reveals a distribution that is uniform instead of being bell-shaped.

8
The worlds smallest mammal is the bumblebee bat, also known as the Kitti's hog-nosed bat. Such bats are roughly the size of a large bumblebee. Listed below are weights from a sample of these bats. Assuming that the weights of all such bats have a standard deviation of 0.30g use a 0.05 significance level to test the claim that these bats are from the same population with a known mean of 1.8g. Do the bats appear to come from the same population?

1.7 1.6 1.5 2.0 2.3 1.6 1.6 1.8 1.5 1.7 2.2 1.4 1.6 1.6 1.6

9
The world's smallest mammal is the bumblebee bat, also known as the Kitti's hog nosed bat. Such bats are roughly the size of a large bumblebee. Listed below are weights from a sample of these bats. Using a 0.05 significance level, test the claim that these weights come from a population with a standard deviation equal to 0.30 g, which is the standard deviation of weights of the bumblebee bats from one region in Thailand. Do these bats appear to have weights with the sam variation as the bats from that region in Thailand?

1.7 1.6 1.5 2.0 2.3 1.6 1.6 1.8 1.5 1.7 2.2 1.4 1.6 1.6 1.6

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