Let x be a random variable that represents the weights in kilograms (kg) of healthy adult female deer (does) in December in a national park. Then x has a distribution that is approximately normal with mean μ = 52.0 kg and standard deviation σ = 9.0 kg. Suppose a doe that weighs less than 43 kg is considered undernourished.
(a) What is the probability that a single doe captured (weighed and released) at random in December is undernourished? (Round your answer to four decimal places.)

(b) If the park has about 2850 does, what number do you expect to be undernourished in December? (Round your answer to the nearest whole number.)

(c) To estimate the health of the December doe population, park rangers use the rule that the average weight of n = 80 does should be more than 49 kg. If the average weight is less than 49 kg, it is thought that the entire population of does might be undernourished. What is the probability that the average weight x for a random sample of 80 does is less than 49 kg (assuming a healthy population)? (Round your answer to four decimal places.)

(d) Compute the probability that

x < 53.1 kg for 80 does (assume a healthy population). (Round your answer to four decimal places.)

Suppose park rangers captured, weighed, and released 80 does in December, and the average weight was

x= 53.1 kg. Do you think the doe population is undernourished or not? Explain.

60 % of the chips used by a computer manufacturing company (CMC) are provided by supplier A and the rest by supplier B. The two suppliers have the following faulty chip rates: 0.3% for supplier A and 0.8% for supplier B. Consider a random batch of 20 chips (chips cannot be distinguished by supplier). a. Show that the expected number of faulty chips is 0.1. [15 marks] b. What is the probability of having two or more faulty chips? Justify your answer and the choice of the probability distribution. [15 marks] If the daily production of computer is 1000, and each computer has one chip: c. What is the probability that 8 or more computers do not pass the final quality check due to faulty chips? Justify your approach. [20 marks] The company, CMC, has to decide whether to implement a quality control screening of chips when delivered by the suppliers. In this case, only the chips that pass the screening will be used in the production process. Previous experience of quality control screenings reported that in 95% of the cases it was able to correctly identify faulty chips and in 3% of the cases was giving false negatives (meaning that even though the chip was working correctly the screening identified it as faulty). The screening process will be convenient only if the average number of daily faulty computers will be smaller than 3. d. Should the company implement the quality control screening? [20 marks]

60 % of the chips used by a computer manufacturing company (CMC) are provided by supplier A and the rest by supplier B. The two suppliers have the following faulty chip rates: 0.3% for supplier A and 0.8% for supplier B. Consider a random batch of 20 chips (chips cannot be distinguished by supplier). a. Show that the expected number of faulty chips is 0.1. [15 marks] b. What is the probability of having two or more faulty chips? Justify your answer and the choice of the probability distribution. [15 marks] If the daily production of computer is 1000, and each computer has one chip: c. What is the probability that 8 or more computers do not pass the final quality check due to faulty chips? Justify your approach. [20 marks] The company, CMC, has to decide whether to implement a quality control screening of chips when delivered by the suppliers. In this case, only the chips that pass the screening will be used in the production process. Previous experience of quality control screenings reported that in 95% of the cases it was able to correctly identify faulty chips and in 3% of the cases was giving false negatives (meaning that even though the chip was working correctly the screening identified it as faulty). The screening process will be convenient only if the average number of daily faulty computers will be smaller than 3. d. Should the company implement the quality control screening? [20 marks]

A bag contains 9 9 w h i t e w h i t e marbles, 3 3 y e l l o w y e l l o w marbles, 7 7 b l u e b l u e marbles. If one marble is drawn from the bag then replaced, what is the probability of drawing a w h i t e w h i t e marble then a b l u e b l u e marble? In a number guessing game. You ask a person to guess a number from one 1 to 10. If the person makes a random guess, what is the probability their guess will be less than 5 5 ? A bag contains 3 3 r e d r e d marbles, 7 7 b l u e b l u e marbles, 9 9 y e l l o w y e l l o w marbles. If one marble is drawn from the bag but not replaced, what is the probability of drawing a r e d r e d marble then a y e l l o w y e l l o w marble? Add Work

A researcher wants to investigate why some individuals released from prison on parole reoffend whereas others do not. As a starting point, the researcher considers the following probit model: Pr(??=1)=Φ(?0+?1??+?2ln??⁡+?3(??×ln??)), (2.1) where Φ denotes the standard normal cumulative distribution function, ln denotes the natural log and: ??=1 if paroled individual ? reoffends within three years of being paroled (0 otherwise); ??=1 if individual ? is male (0 otherwise); ?? is the number of years since individual ? was paroled. The above model was estimated using data on a sample of convicted individuals who were recently released from prison on parole, whose reoffending behavior was subsequently monitored. The (rounded) parameter estimates obtained (with standard errors in brackets) were ?̂0=0.00 (0.12), ?̂1=0.7 (0.02), ?̂2=−0.5 (0.07),⁡?̂3=−0.5 (0.10). (a) Robin Banks was paroled a year ago. What is his estimated probability of reoffending? (b) Emma Besler (everyone calls her ‘Em’) has an estimated probability of reoffending of 50%. How long ago was she paroled? (c) Based on a hypothesis test at the 5% significance level, could Em’s probability of reoffending be as high as 60%? (d) Estimate the number of years a man must be on parole to be equally likely to re-offend as a woman paroled 12 months ago.

Let x be a random variable that represents the weights in kilograms (kg) of healthy adult female deer (does) in December in a national park. Then x has a distribution that is approximately normal with mean μ = 64.0 kg and standard deviation σ = 7.7 kg. Suppose a doe that weighs less than 55 kg is considered undernourished.

(a) What is the probability that a single doe captured (weighed and released) at random in December is undernourished? (Round your answer to four decimal places.) Incorrect: Your answer is incorrect.

(b) If the park has about 2250 does, what number do you expect to be undernourished in December? (Round your answer to the nearest whole number.) Incorrect: Your answer is incorrect. does

(c) To estimate the health of the December doe population, park rangers use the rule that the average weight of n = 55 does should be more than 61 kg. If the average weight is less than 61 kg, it is thought that the entire population of does might be undernourished. What is the probability that the average weight x for a random sample of 55 does is less than 61 kg (assuming a healthy population)? (Round your answer to four decimal places.) Incorrect: Your answer is incorrect.

(d) Compute the probability that x < 65.1 kg for 55 does (assume a healthy population). (Round your answer to four decimal places.)

Part IV: Finding exact probabilities. FOR EACH, DRAW A PICTURE AND USE THE Z-TABLE ON CARMEN.

20. The Z-table always gives you the probability of being between ________ and  the number you are looking up.

21. The number you are looking up should have ______digit(s) before the decimal point and _____ digit(s) after the decimal point.

22. Suppose the Z value is 2.00. Which row and column do you look in to find P(0< Z < 2.00)?

23. How do you find P(Z < 2.00)? (Note you have to do this in 2 parts. Hint: What is the probability that Z is less than 0? Use that as one of the parts)

24. How do you find P(Z > 2.00)? (Note the table does not have “>” probabilities. If half of the probability is greater than 0, how much of it must be greater than 2? Draw a picture. )

25. a. What is P(-1.26  < Z < 0)?  (Note the Z table has no negative values. Use SYMMETRY to do this.)

25.  b.  Find P(Z > -1.26)

25.  c. Find P(Z < -1.26)

26.a   Now find P(-1 < Z < 2). Do this in two parts and sum them together. Use symmetry to get the left part.

26b. Find P(1<Z<2)

Suppose that a drawer contains 8 marbles: 2 are red, 2 are blue, 2 are green, and 2 are yellow. The marbles are rolling around in a drawer, so that all possibilities are equally likely when they are drawn. Alice chooses 2 marbles without replacement, and then Bob also chooses 2 marbles without replacement. Let Y denote the number of red marbles that Alice gets, and let X denote the number of red marbles that Bob gets.

a. Find probability mass function for Y, i.e., for the number of red marbles that Alice gets, i.e., find pY(y) for y=0,1,2.

pY(0)=

pY(1)=

pY(2)=

b. Find E(Y).

.researcher suspected that the number of between meal snacks eaten by students in a day during final examinations
might depend on the number of tests a student had to take on that day. The accompanying table shows joint probabilities,
estimated from a survey.
Number of
No of snacks Y No of test
0 1 2 3
0
1 0.07 0.09 0.06 0.01
2   0.07 0.06 0.07 0.01
3   0.06 0.07 0.14 0.03
4   0.02 0.04 0.16 0.04
.
I. . Find probability of mean and variance of 3X-2Y
II. . Find the covariance between X and Y.
B. Find mean and variance of distribution by using M.G.F technique.(1− ?)^×(p)

A government agency counselling customers (citizens) subjected to domestic violence employs four lawyers. The agency manages complaints from three types of customers. The time a lawyer spends with each type of customers is exponentially distributed, with a mean of 15 minutes. Inter-arrival times for each customer type are exponential, with the average number of arrivals per hour for each customer type being observed as follows: type 1, 3 customers per hour; type 2, 5 customers per hour; and type 3, 3 customers per hour. Assume that type 1 customers have the highest priority, and type 3 customers have the lowest priority. Preemption is not allowed. a) What is the average length of time that each type of customers must wait before seeing a lawyer?