1Consider the following observations on shear strength of a joint bonded in a pa

Question

1Consider the following observations on shear strength of a joint bonded in a particular manner: 30.0 44.0 33.1 66.7 81.5 22.2 40.4 16.4 73.7 36.6 109.9

a)Find sample mean (2 points),

b)Find sample variance (3 points)

c) Find sample median (2 points).

d) Construct a boxplot. Do you have outliers? Comment on the plot (3 points )

2. At a certain gas station, 40% of the customers use regular unleaded gas, 35% use extra unleaded gas), and 25% use premium unleaded gas. Of those customers using regular gas, only 30% fill their tanks (event B). Of those customers using extra gas, 60% fill their tanks, whereas of those using premium, 50% fill their tanks.

a) What is the probability that the next customer will request extra unleaded gas and fill the tank?(2 POINTS)

b) What is the probability that the next customer fills the tank? (4 POINTS)

c) If the next customer fills the tank, what is the probability that regular gas is requested? Extra gas? (4 POINTS)

3.A college professor always finishes his lectures within 2 minutes after the bell rings to end the period and the end of the lecture. Let X = the time that elapses between the bell and the end of the lecture and suppose the pdf of X is f(x) = cx2 0 < x < 2 0 otherwise

a) Findc.(3 points)

b)What is the probability that the lecture continues beyond the bell for between 60 and 90 seconds?(3 points)

c) Find an expected value of X .(4 points)

4. Twenty-five percent of all telephones of a certain type are submitted for service while under warranty. Of these, 60% can be repaired whereas the other 40% must be replaced with new units. If a company purchases ten of these telephones, what is the probability that exactly two will end up being replaced under warranty?(5 points)

5.The number of parking tickets issued in Grand Rapids on any given weekday has a Poisson distribution with parameter = 50. What is the approximate probability that between 40 and 70 tickets are given out on a particular day? (Hint: When is large, a Poisson random variable has approximately a normal distribution.) (5 points)

6. Assuming that the observations in problem one are normally distributed estimate a 95% confidence interval for the population mean of shear strength of a joint bonded in a particular manner. (10 points)

7. A random sample of 150 recent donations at a certain blood bank reveals that 82 were type A blood. Does this suggest that the actual percentage of type A donations differs from 40%, the percentage of the population having type A blood? Carry out a test of the appropriate hypotheses using a significance level of .01. Would your conclusion have been different if a significance level of .05 had been used? (10 points).

More problems:

1. Let A denote the event that the next item checked out at a college library is a math book, and let B be the event that the next item checked out is a history book. Suppose that P(A) = .40 and P(B) = .50.

a. Why is it not the case thatP(A) + P(B) =1?

b. Calculate P(A0 )

c. Calculate P(A B).

d. Calculate P(A0 B0 ).

2. A real estate agent is showing homes to a prospective buyer. There are ten homes in the desired price range listed in the area. The buyer has time to visit only four of them.

a. In how many ways could the four homes be chosen if the order of visiting is considered?

b. In how many ways could the four homes be chosen if the order is disregarded?

c. If four of the homes are new and six have previously been occupied and if the four homes to visit are randomly chosen, what is the probability that all four are new? (The same answer results regardless of whether order is considered.)

3. A geologist has collected 10 specimens of basaltic rock and 10 specimens of granite. The geologist instructs a laboratory assistant to randomly select 15 of the specimens for analysis.

a. What is the pmf of the number of granite specimens selected for analysis?

b. What is the probability that all specimens of one of the two types of rock are selected for analysis?

c. What is the probability that the number of granite specimens selected for analysis is within 1 standard deviation of its mean value?

4. An insurance company offers its policyholders a number of different payment options. For a randomly selected policyholder, let X = the number of months between successive payments. The cdf of X is as follows: F(x) = 0 x < 1 0.3 1 x < 3 0.4 3 x < 4 0.45 4 x < 6 0.6 6 x < 12 1 x 12

a. What is the pmf of X?

b. Using just the cdf, compute P(3 X 6)

c. Using just the pmf, compute P(X > 6).

5. The cdf of checkout duration X for a book on a 2-hour reserve at a college library is given by:

F(x) =

0 x < 0

x2 /4 0 x < 2

1 x 2

a) Using cdf Find P(0.5 X 1).

b) Find pdf of X. c) Find E(X)

6. The distribution of resistance for resistors of a certain type is known to be normal, with 10% of all resistors having a resistance exceeding 10.634 ohms, and 5% having a resistance smaller than 9.7565 ohms. What are the mean value and standard deviation of the resistance distribution?

7. Let X have a uniform distribution on the interval [0, 3].

a. Find the median the X.

b. Compute E(X), V(X) .

8. A CI is desired for the true average stray-load loss (watts) for a certain type of induction motor when the line current is held at 10 amps for a speed of 1500 rpm. Assume that strayload loss is normally distributed with = 3.0.

a. Compute a 95% CI for µ when n = 25 and ¯x = 60.

b. Compute a 88% CI for when n = 100 and ¯x = 60.

9. A sample of 14 joint specimens of a particular type gave a sample mean proportional limit stress of 8.50 MPa and a sample standard deviation of .80 MPa. Calculate and interpret a 95% lower confidence bound for the true average proportional limit stress of all such joints. What, if any, assumptions did you make about the distribution of proportional limit stress?

10. A random sample of 100 observations produced a sample proportion of .25. Find 90% confidence interval for the population proportion p.

11.A sample of 12 radon detectors of a certain type was selected, and each was exposed to 100 pCi/L of radon. The resulting readings were as follows: 104.3 89.6 89.9 95.6 95.2 90.0 98.8 103.7 98.3 106.4 102.0 91.1 Does this data suggest that the population mean reading under these conditions differs from 100? State and test the appropriate hypotheses using = .05

Explanation / Answer

1)
a) Sample mean = 50.409
b) Sample variance = summation of ( x - mean)^2 / n
= 831.4129
c) Sample median = 40.4

2)
A1 = {use regular unleaded gas} P(A1) = 0.40 P(B|A1) = 0.30
A2 = {use extra unleaded gas} P(A2) = 0.35 P(B|A2) = 0.60
A3 = {use premium unleaded gas} P(A3) = 0.25 P(B|A3) = 0.50
B = {fill the tank completely}
B’ = {not filling the tank completely}

a) P(A2B) is found using the definition of conditional probability.
P(A2 B) = P(B A2) = P(B | A2)P(A2)
= (0.60)(0.35) = 0.21

b)
P(B) is found by using the law of total probability.
P(B) = P(B A1) + P(B A2)+P(B A3)
P(B) = P(B|A1)P(A1) + P(B|A2)P(A2)+P(B|A3)P(A3)
P(B) = (0.30)(0.40) + (0.60)(0.35) + (0.50)(0.25)
= 0.455
C)
P(A1B) is found using the definition of conditional probability.

P(A1|B) = P(A1 B) / P(B) = P(B A1) / P(B) = P(B|A1)P(A1)/P(B) = (0.30)*(0.40)/ 0.455 = 0.264

P(A2|B) = P(A2 B) / P(B) = P(B A2) / P(B) = P(B|A2)P(A2)/P(B) = (0.60)*(0.35)/ 0.455 = 0.461

P(A3|B) = P(A3 B) / P(B) = P(B A3) / P(B) = P(B|A3)P(A3)/P(B) = (0.50)*(0.25)/ 0.455 = 0.275

4)
Let S represent a telephone that is submitted for service while under warranty and must
be replaced. Then p = P(S) = P(replaced | submitted)× P(submitted) = (.40)(.25) = .10.
Thus, X, the number among the company’s 10 phones that must be replaced, has a
binomial distribution with n = 10, P = 0.10.
  

Therefore P(2)= P( X= 2) = 10C2 * (.10)^2 (.90)^8
= .1937

10)

N = 100 , P = 0.25
Z VALUE AT 90% = 1.645
CI = p +/- z * sqrt(p * ( 1 - p) / n)
= 0.25 + /- 1.645 * sqrt(0.25 * 0.75 / 100)
= (0.1787 , 0.3212)

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