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)

2. Industrialist H.E. Pennypacker wants information on the customers that patronize his bicycle stores. He surveyed 81 randomly-selected individuals who made a purchase at his Pasadena store to find out how much they spent, on average. The mean amount spent was $62 with a standard deviation of $15.

a) What is the population in this study?

b) Construct a 95% CI for the mean amount of money an individual spends at the Pasadena store. Explain the meaning of this CI (i.e., what does it say about the parameter of interest?).

c) Construct a 99% CI for the mean amount of money spent at the Pasadena store. Contrast this interval with the one from b and explain why it is different.

d) Suppose 100 people were initially surveyed, but 19 of them actually refused to answer (leading to the final sample size of 81). If the 19 individuals who refused to answer spent considerably less money than the 81 who did respond, how would this affect the estimate of the mean and the CI?

1.Which of the following is not a characteristic of a binomial random variable?

a. n identical trials

b. probability of failure is the same for each trial

c. non-correlated outcomes for each trial

d. non of the above

2. Which of the following are not binomial random variables(multiple answers)

a. one hundred randomly selected individuals are asked about their opinion on health insurance

b. one hundred people at a bar are asked about whether they are for or against restricting the sale of alcohol

c. one hundred randomly selected people are asked if they are in favor of a single payer health care system

d. one hundred randomly selected individuals are asked abut their marital status

3. Supposed that the probability of a randomly selected individual developing side effects from a new diet is 20%. If ten subjects are testing the diet, what is the probability that exactly 3 individuals develop side effects? (enter your answer as follows: 10.1%)

4. Suppose that the probability of a randomly selected individual developing side effects from a new diet is 20%. If ten subjects are testing the diet, what is the probability that at most 3 individuals develop side effects? (enter your answer as follows: 10.1%)

5. Suppose that the probability of a randomly selected individual developing side effects from a new diet is 20%. What is the expected number of subjects that would develop side effects if 500 individuals tested the diet?

6. Suppose that the probability of a randomly selected individual developing side effects from a new diet is 20%. Would it be unusual if only 35 out of 400 individuals trying the diet developed side effects?

a. yes, since the probability of 35 cases out of 400 is less than 1%

b. yes, since the 35 cases are more than two standard deviation from the mean

c. all of the above

d. none of the above

Lets use Excel to simulate rolling two 8-sided dice and finding the rolled sum.

• Open a new Excel document.

• Click on cell A1, then click on the function icon fx and select Math&Trig, then select RANDBETWEEN.

• In the dialog box, enter 1 for bottom and enter 8 for top.

• After getting the random number in the first cell, click and hold down the mouse button to drag the lower right corner of this first cell, and pull it down the column until 25 cells are highlighted. When you release the mouse button, all 25 random numbers should be present.

• Repeat these four steps for the second column, starting in cell B1.

• Put the rolled sum of two dice in the third column: Highlight the first two cells in the first row and click on AutoSum icon. Once you receive the sum of two values in the third cell, drag the lower right corner of this cell, C1, down to C25. This will copy the formula for all 25 rows. We now have 25 trials of our experiment.

Once these steps are completed, attach a screenshot of your Excel file to your assignment.

(a) Find the theoretical probability that the rolled sum of both dice is 8.

(b) Based on the results of our experiment of 25 trials, obtain the relative frequency approximation to the probability found in (a). You can do so in Excel in two different ways: i) create the histogram of the third column data, then scroll the mouse over the relevant bar - this will give you the frequency with which you can determine the relative frequency; or ii) in a cell, type the function COUNTIF(C1:C25,8)

(c) Generate the frequency distribution histogram of your experiment of 25 trials, and copy it to a Word document. Make sure to add a title to your histogram.

(d) Repeat the simulation for 100 and 1000 trials, and calculate the relative frequency for each, and create the frequency distribution histogram - resize the 3 histograms so that all 3 fit beside each other in a row.

(e) Identify which of the 3 relative frequencies for ’8’ is the closest value to the theoretical probability found in (a). Briefly explain how these experiments demonstrate the Law of Large Numbers.

(f) Identify the shape of the probability distribution (uniform, bell-curved, right-skewed or left-skewed).

Suppose you are a researcher in a hospital. You are experimenting with a new tranquilizer. You collect data from a random sample of 9 patients. The period of effectiveness of the tranquilizer for each patient (in hours) is as follows: 2 2.4 2.7 2.4 2.1 2.1 2.2 2.9 2.1

What is a point estimate for the population mean length of time. (Round answer to 4 decimal places)

Which distribution should you use for this problem? normal distribution or t-distribution

What must be true in order to construct a confidence interval in this situation?

The population standard deviation must be known

The population must be approximately normal

The population mean must be known

The sample size must be greater than 30

Construct a 90% confidence interval for the population mean length of time. Enter your answer as an open-interval (i.e., parentheses) Round upper and lower bounds to two decimal places

What does it mean to be "90% confident" in this problem?

There is a 90% chance that the confidence interval contains the population mean

The confidence interval contains 90% of all samples 90% of all simple random samples of size 9 from this population will result in confidence intervals that contain the population mean

Suppose that the company releases a statement that the mean time for all patients is 2 hours.

Is this possible? No Yes

Is it likely? Yes No

3. A box contains 5 red balls, 3 blue balls and 1 black balls. Take two balls out randomly. Let X be number of red balls and Y be the number of black balls.

(1) Find the joint distribution of (X, Y ).

(2) Find P(X = 1|Y = 1).