Use R to generate n = 400 samples (idependent identically distributed random numbers) of X ∼ N(0, 4). For each Xi , simulate Yi according to Yi = 3 + 2.5Xi + εi , where εi ∼ N(0, 16), i = 1, ..., n. Use R to solve the following questions. (a) Compute the least square estimators of βˆ 0 and βˆ 1. (b) Draw a regression line according to the numbers computed in (a). Plot Y and X with this regression line. Hint: Use rnorm.

A coin with probability p>0 of turning up heads is tossed 4 times. Let X be the number of times heads are tossed.

(a) Find the probability function of X in terms of p.

(b) The result above can be extended to the case of n independent tosses (that is, for a generic number of tosses), and the probability function in this case receives a very specific name. Find the name of this particular probability function.

Notice that the probability of turning up tails in one toss is 1−p.

a)  A coin is flipped 6 times, find the probability of getting exactly 4 heads.  Hint: The Binomial Distribution Table can be very helpful on questions 19-21.  If you use the table for this question, give your answer exactly as it appears.  If you calculated your answer, round to the thousandths place.

b) A coin is flipped 6 times. Find the probability of getting at least 3 heads. If you used a table to help find your answer, give it to the thousandths place. If you used a formula to calculate your answer, round to the thousandths place.

c) A coin is flipped 6 times, find the probability of getting at most 2 heads. If you used a table to help find your answer, give it to the thousandths place. If you used a formula to calculate your answer, round to the thousandths place.

Suppose a road is flooded with probability p = 0.10 during a year and not more than one flood occurs during a year.

What is the probability that it will be flooded at most twice during a 10-year period?

After a group of (volunteer) experimental subjects achieved a specified blood alcohol level from consuming liquor, psychologists placed each subject in a room and threatened them with electric shocks. Using special equipment, the psychologists recorded the startle response time (in milliseconds) of each subject. The mean and standard deviation of the startle response times is 37.9 milliseconds and 12.4 milliseconds, respectively.

a. Without knowing the shape of the distribution of these response times, what is the interval that contains at least 75% of response times?

b. If you are told that the shape of the distribution of response times is normal,

i. What interval contains approximately 68% of startle response times?

ii. What interval contains approximately 95% of startle response times?

iii. What interval contains 99.7% (i.e., almost all) of startle response times?

iv. Approximately what percentage of startle response times fall below 25.5 milliseconds? HINT: Use the fact that a normal distribution is symmetric around the mean to answer this question.

Suppose the incidence of ketosis in Holstein dairy cows is 11.8% in the state of Michigan (Dyk and Emery, 1996; Proc Tri-State Nutrition Conference). Assume the state of ketosis (1 = diseased, 0 = not-diseased) is independent between all animals; i.e., the occurrence or non-occurence of ketosis on any one animal is in no way related to whether ketosis occurs in any other animal. Furthermore, suppose that there are a large number of herds in the state for which the probability of ketosis (11.8%) was the same for each herd.

A)Within any one herd of size 60, what is the exact probability of having 4 or less cows being afflicted with ketosis.

B)Redo (a) but use a normal approximation to determine this probability. Do this with and without the continuity adjustment. Are either of these results consistent with the exact probability you provided in your answer to a)?

4. A space probe encounters a planet capable of sustaining life on average every 3.4 lightyears. (Recall that a lightyear is a measure of distance, not time.)

a) Let L be the number of life-sustaining planets that the probe encounters in 10 lightyears. What are the distribution, parameter(s), and support of L?

b) What is the probability that the probe encounters at least 2 life-sustaining planets in 10 lightyears?

c) The probe has just encountered a life-sustaining planet. What is the probability that it takes more than 4 lightyears to encounter the next life-sustaining planet? What distribution and parameter(s) are you using?

d) Suppose the probe has not encountered a life-sustaining planet for 2.5 lightyears. Knowing this, what is the probability that it will take at most 8 lightyears to detect the next life-sustaining planet?

e) The probe has encountered 10 life-sustaining planets in the last 25 lightyears. What is the probability that there are 3 life-sustaining planets in the first 5 lightyears of this 25-lightyear span?

what are the degrees of freedom?

a. yes, at p < .05

b. yes, at p < .01

c. yes, at p < .001

d. no, at p > .05

Let’s check in on your tomato plants…again. Say that you have planted 240 tomato seeds. You set up an infrared camera and record their germination over 24 hours.

Do these data meet the assumptions for the x² goodness-of-fit test? Explain.

What are the degrees of freedom (df) for these data?

Use the sample information

x¯= 40, σ = 7, n = 13 to calculate the following confidence intervals for μ assuming the sample is from a normal population.

(a) 90 percent confidence. (Round your answers to 4 decimal places.)
  
The 90% confidence interval is from ___ to___

(b) 95 percent confidence. (Round your answers to 4 decimal places.)
  
The 95% confidence interval is from ___ to____

(c) 99 percent confidence. (Round your answers to 4 decimal places.)
  
The 99% confidence interval is from ____to____

(d) Describe how the intervals change as you increase the confidence level.
  

1)The interval gets narrower as the confidence level increases.

2)The interval gets wider as the confidence level decreases.

3)The interval gets wider as the confidence level increases.

4)The interval stays the same as the confidence level increases.

Communicating the correct amount of data to stakeholders is important. Discuss specific strategies that can be used to ensure the right balance in terms of communicating dataset findings to stakeholders. Illustrate your ideas with specific examples.

For this problem, carry at least four digits after the decimal in your calculations. Answers may vary slightly due to rounding.

In a random sample of 520 judges, it was found that 291 were introverts.

(a) Let p represent the proportion of all judges who are introverts. Find a point estimate for p. (Round your answer to four decimal places.)


(b) Find a 99% confidence interval for p. (Round your answers to two decimal places.)

Give a brief interpretation of the meaning of the confidence interval you have found.

We are 99% confident that the true proportion of judges who are introverts falls outside this interval.We are 1% confident that the true proportion of judges who are introverts falls above this interval.    We are 99% confident that the true proportion of judges who are introverts falls within this interval.We are 1% confident that the true proportion of judges who are introverts falls within this interval.

(c) Do you think the conditions np > 5 and nq > 5 are satisfied in this problem? Explain why this would be an important consideration.

Yes, the conditions are satisfied. This is important because it allows us to say that p̂ is approximately binomial.Yes, the conditions are satisfied. This is important because it allows us to say that p̂ is approximately normal.    No, the conditions are not satisfied. This is important because it allows us to say that p̂ is approximately binomial.No, the conditions are not satisfied. This is important because it allows us to say that p̂ is approximately normal.

The average LSAT score (the standardized test required to apply to law school) in the United States is µ =150 (σ = 10). Also, the LSAT is normally distributed. Use these parameters to answer the following questions:

• If someone took an LSAT test and received a 153, what proportion of scores will be less than this?
• If someone took an LSAT test and received a 143, what proportion of scores will be greater than this?
• What proportion of LSAT scores will be within the interval of 138 to 172?
• What proportion of LSAT scores will be outside the interval of 125 to 175?
• If someone wants to have an LSAT score higher than 90% of all other test-takers, what score do they need to earn?

Box 1 contains 3 red balls, 5 green balls and 2 white balls. Box 2 contains 5 red balls, 3 green balls and 1 white ball. One ball of unknown color is transferred from Box 1 to Box 2. (a) What is the probability that a ball drawn at random from Box 2 is green? (b) What is the probability that a ball drawn from Box 1 is not white?

Roll two fair dice. Each die has six faces.

A. Let A be the event that either a 3 or 4 is rolled first followed by an odd number. P(A) =  Round your answer to two decimal places.

B. Let B be the event that the sum of the two dice is at most 7. P(B) =  Round your answer to two decimal places.

C. Are A and B mutually exclusive events? (Yes or No)

D. Are A and B independent or dependent events? (Independent or Dependent)