4. You are visiting the rainforest, but unfortunately your insect repellent has run out. As a result, at each second, a mosquito lands on your neck with probability 0.5. If one lands, with probability 0.2 it bites you, and with probability 0.8 it never bothers you, independently of other mosquitoes.
a. What is the expected time between successive mosquito bites? What is the variance of the time between successive mosquito bites?
b. In addition, a tick lands on your neck with probability 0.1. If one lands, with probability 0.7 it bites you, and with probability 0.3, it never bothers you, independently of other ticks and mosquitoes. Now, what is expected time between successive bug bites? What is the variance of the time between successive bug bites?

Box A contains 6 red balls and 3 green balls, whereas box B contains 3 red ball and 15 green balls.

Stage one. One box is selected at random in such a way that box A is selected with probability 1/5 and box B is selected with probability 4/5.

Stage two. Finally, suppose that two balls are selected at random with replacement from the box selected at stage one.

g) What is the probability that both balls are red?

h) Given that both balls are red, what is the probability they came from box A?

i) What is the probability that one ball is red and the other is green?

j) Given that one ball is red and the other is green, what is the probability they came from box A?

Tyler lives in Anchorage and has loss averse preferences. In particular, Tyler values a gain of amount x as u(x) = x^(1/2) and values a loss of −x as u(−x) = −2x 1 2

(a) What is the maximum amount of money that Tyler would pay for a lottery that pays $1000 with probability 1/2 and $0 with probability 1/2 ?

(b) What is the maximum amount of money that Tyler would pay to avoid playing a lottery that loses $1000 with probability 1/2 and loses $0 with probability 1/ 2 ?

(c) What is the maximum amount of money that Tyler would pay to avoid playing a lottery that loses $1000 with probability 1/2 and gains $1000 with probability 1/2 ?

1. CPK and SGOT tests are used in the diagnosis of myocardial infarction (MI). When the CPK test is given to a patient who does not have a MI, the probability of a negative finding (i.e. its specificity) is 0.6. The probability that the SGOT test will be negative for a non-MI patient is 0.7. When both tests are given to a non-MI patient the probability that at least one is negative is 0.9. For a non-MI patient who has both tests:

1. What is the probability that both tests are negative?
2. What is the probability that both tests are positive?

Hints: (1) Answer is not 0.12 -- tests are not to be assumed to be independent.

           (2) Using 2-by-2 table to structure your calculations can help.

3. What is the probability that at least one test is positive?

Lab - Chapter 5 - Normal Distribution Problems

1. The machine that packages 5 lb. bags of sugar is designed to put an average (mean) of 5.1 lbs. with a standard deviation of 0.4 lbs. into the package.

a. What is the probability that a bag of sugar weighs less than 5 lbs.?

b. What is the probability that a bag of sugar weighs more than 5.1 lbs.?

c. What is the probability that a bag weighs between 4.5 and 5.5 lbs.?

d. What is the probability that a bag weighs between 5.4 and 6.6 lbs.?

2. A bag of individually wrapped candy claims to contain 45 pieces. The machine packaging the candy is designed to put an average of 43 pieces in the bag with a standard deviation of 4 pieces.

a. What is the probability that the bag has more than 45 pieces?

b. What is the probability that the bag has more than 51 pieces?

c. What is the probability that the bag contains between 43 and 47 pieces of candy?

d. What is the probability that the bag contains between 47 and 53 pieces of candy?

3. In recent years, the results of a particular college entrance exam showed an average score of 55 with a standard deviation of 4 points.

a. Based on these results, what is the probability that an incoming student scores below a 50?

b. What is the probability that a student scores above a 60?

c. What is the probability that a student scores between a 53 and a 59?

d. What is the probability that a student scores below a 58?

4. A bag of potato chips claims to contain 7 oz. of potato chips. Random sampling determines that the bags contain an average of 7.2 oz. with a standard deviation of 0.081 oz.

a. What is the maximum weight that 20% of the bags contain less than?

b. What is the minimum weight that the heaviest 15% of the bags contain?

c. In what weight interval are the middle 60% of the bags contained?

Decide whether you can use the normal distribution to approximate the binomial distribution. If you​ can, use the normal distribution to approximate the indicated probabilities and sketch their graphs. If you​ cannot, explain why and use the binomial distribution to find the indicated probabilities. Five percent of workers in a city use public transportation to get to work. You randomly select 269 workers and ask them if they use public transportation to get to work.

(Complete parts A through D)

Can the normal distribution be used to approximate the binomial​ distribution?

a.​Yes, because both np ≥ 5 and nq ≥ 5.

b.​No, because np < 5.

c.​No, because nq <5.

A) ​- Find the probability that exactly 20 workers will say yes.

What is the indicated​ probability? (____) Round to four decimal places as​ needed.

Sketch the graph of the normal distribution with the indicated probability shaded.

B) ​- Find the probability that at least 7 workers will say yes.

What is the indicated​ probability? (____) Round to four decimal places as​ needed.

Sketch the graph of the normal distribution with the indicated probability shaded.

C) - Find the probability that fewer than 20 workers will say yes.

What is the indicated​ probability? (____) Round to four decimal places as​ needed.

Sketch the graph of the normal distribution with the indicated probability shaded.

D) - A transit authority offers discount rates to companies that have at least 30 employees who use public transportation to get to work. There are 452 employees in a company. What is the probability that the company will not get the​ discount?

Can the normal distribution be used to approximate the binomial​ distribution?

a.​ No, because nq < 5.

b.​ No, because np < 5.

c.​ Yes, because both np ≥ 5 and nq ≥ 5.

What is the probability that the company will not get the​ discount? (____) Round to four decimal places as​ needed.

Sketch the graph of the normal distribution with the indicated probability shaded.

Suppose a geyser has a mean time between eruptions of 93 minutes. If the interval of time between the eruptions is normally distributed with standard deviation 28 minutes , answer the following questions.

(a) What is the probability that a randomly selected time interval between eruptions is longer than 107 minutes?

(b) What is the probability that a random sample of 7 time intervals between eruptions has a mean longer than 107 minutes?

(c) What is the probability that a random sample of 21 time intervals between eruptions has a mean longer than 107 minutes?

(d) What effect does increasing the sample size have on the probability? Provide an explanation for this result. Choose the correct answer below.

A. The probability increases because the variability in the sample mean increases as the sample size increases.

B. The probability decreases because the variability in the sample mean increases as the sample size increases.

C. The probability increases because the variability in the sample mean decreases as the sample size increases.

D. The probability decreases because the variability in the sample mean decreases as the sample size increases.

(e) What might you conclude if a random sample of 21 time intervals between eruptions has a mean longer than 107 minutes? Choose the best answer below.

A. The population mean may be greater than 93.The population mean may be greater than 93.

B. The population mean must be more than 93, since the probability is so low. The population mean must be more than 93, since the probability is so low.

C. The population mean must be less than 93, since the probability is so low. The population mean must be less than 93, since the probability is so low.

D. The population mean is 93 minutes, and this is an example of a typical sampling.

Exhibit 2

Current statistics show that, about 5% of the patients who are infected with the Novel Coronavirus are in serious or critical condition, and need ventilators and oxygen facilities. Suppose this is the population proportion. (The questions related to this Exhibit are designed so that you can see how statistical analyses can be used to fight against pandemics.)

Question 7

Refer to Exhibit 2. Assume that in the City of Gotham, in the first week of outbreak, 256 citizens are tested positive for COVID-19. At least how many ventilators should be prepared to meet the possible demand. (Round up to the nearest integer that is larger than the result.)

Question 8

Refer to Exhibit 2. Suppose the No. 1 district of the City of Gotham has 8100 residents living in it. If you were the head of the health department of the No. 1 district. To cope with the possible all-infection outbreak of COVID-19 in your district, you prepared 600 ventilators. Assuming in the worst case scenario, what is the probability that your prepared medical equipments are overwhelmed by the serious conditioned patients who are in need of the ventilators? (Round up to nearest four decimal place.)

Question 9

Refer to Exhibit 2. In the first week of out break, the total number of confirmed cases in the No. 1 district of Gotham is 96. Out of these 96 cases, no one has developed serious conditions yet. But you want to use the normal approximation method to estimate a probability that, from these 96 infected patients, more than 10 cases develop a serious condition. Are you able to do so? Why or why not? If yes, please provide the probability value you estimated. (In four decimal places.)

Question 10

Refer to Exhibit 2. In the second week, the total number of confirmed cases in the No. 1 district of Gotham increased to 196. You want to use this sample of 196 cases and the normal approximation method to estimate the probability that, from these 196 patients, more than 19 cases develop a serious condition. Are you able to do so? Why or why not? If yes, please provide the probability value you estimated. (In four decimal places.)

Question 11

Refer to Exhibit 2. Suppose that you are the head of the health department in the City of Zion. You DO NOT know the population proportion of seriously conditioned cases among the people who infected with the COVID-19. But now you have a sample of 625 cases who tested positive of COVID-19, out of these 625 patients, 28 developed serious conditions. What is your estimation of the proportion of seriously conditioned patients? (Round to the nearest four decimal place.)

Question 12

Refer to Exhibit 2. Continue from Question 11. Construct a 88% confidence interval (CI) for the proportion you estimated in Question 11. What is the Lower Confidence Limit of your CI? (Round to the nearest four decimal place.)

Question 13

Refer to Exhibit 2. Continue from Question 12. What is the Upper Confidence Limit of your CI? (Round to the nearest four decimal place.)