2.1. The following data are an i.i.d. sample from a Cauchy(θ, 1) distribution: 1.77, -0.23, 2.76, 3.80, 3.47, 56.75, -1.34, 4.24, -2.44, 3.29, 3.71, -2.40, 4.53, -0.07, -1.05, -13.87, -2.53, -1.75, 0.27, 43.21.
a. Graph the log likelihood function. Find the MLE for θ using the Newton–Raphson method. Try all of the following starting points: -11, -1, 0, 1.5, 4, 4.7, 7, 8, and 38. Discuss your results. Is the mean of the data a good starting point?
Please submit the code in R, thanks.

q3. The probability a car salesman sells a car to a customer is 0.05 Assuming the salesmen sees 12 customers in a week, what is the probability he sells less than 2 cars? Write answer using three decimal places

q4. The Jones family was one of the first to come to the U.S. They had 6 children. Assuming that the probability of a child being a girl is .5, find the probability that the Jones family had: at least 2 girls? at most 2 girls?

Suppose, household color TVs are replaced at an average age of μ = 8.4 years after purchase, and the (95% of data) range was from 5.0 to 11.8 years. Thus, the range was 11.8 – 5.0 = 6.8 years. Let x be the age (in years) at which a color TV is replaced. Assume that x has a distribution that is approximately normal. (a) The empirical rule indicates that for a symmetrical and bell-shaped distribution, approximately 95% of the data lies within two standard deviations of the mean. Therefore, a 95% range of data values extending from μ – 2σ to μ + 2σ is often used for "commonly occurring" data values. Note that the interval from μ – 2σ to μ + 2σ is 4σ in length. This leads to a "rule of thumb" for estimating the standard deviation from a 95% range of data values. Estimating the standard deviation For a symmetric, bell-shaped distribution, standard deviation ≈ range 4 ≈ high value – low value 4 where it is estimated that about 95% of the commonly occurring data values fall into this range. Use this "rule of thumb" to approximate the standard deviation of x values, where x is the age (in years) at which a color TV is replaced. (Round your answer to one decimal place.) yrs (b) What is the probability that someone will keep a color TV more than 5 years before replacement? (Round your answer to four decimal places.) (c) What is the probability that someone will keep a color TV fewer than 10 years before replacement? (Round your answer to four decimal places.) (d) Assume that the average life of a color TV is 8.4 years with a standard deviation of 1.7 years before it breaks. Suppose that a company guarantees color TVs and will replace a TV that breaks while under guarantee with a new one. However, the company does not want to replace more than 6% of the TVs under guarantee. For how long should the guarantee be made (rounded to the nearest tenth of a year)? yrs

A clinic developed a diet to impact body mass (fat and muscle). A nutritionist in the clinic hypothesizes that heavier individuals on the diet will predict more body fat. Below are the data for a sample of clients from the clinic. Weight is measured in kilograms (kg) and percentage body fat is estimated through skinfold measurement. What can the nutritionist conclude with α = 0.01?

a.) Compute the statistic selected:

b.) Compute the appropriate test statistic(s) to make a decision about H₀.
(Hint: Make sure to write down the null and alternative hypotheses to help solve the problem.)
Critical value =

Make an interpretation based on the results.

More weight of individuals on the diet significantly predicts more body fat.

The weight of individuals on the diet does not significantly predict body fat.

As reported in "Runner's World" magazine, the times of the finishers in the New York City 10 km run are normally distributed with a mean of 61 minutes and a standard deviation of 9 minutes. Let x denote finishing time for the finishers. a) The distribution of the variable x has mean ___ and standard deviation ___ b) The distribution of the standardized variable z has mean ___ and standard deviation ___ c) The percentage of finishers with times between 35 and 75 minutes is equal to the area under the standard normal curve between ___ and ___ d) The percentage of finishers with times exceeding 88 minutes is equal to the area under the standard normal curve that lies to the ___ of ___

Respond to the following in a minimum of 175 words:

The standard error of the estimate of the mean is represented by the equation: σ√n Discuss what this equation means, using your own words and explain why we use it. Consider how it relates to the fact that we are making assumptions about the population and not just the sample.

An employee of a small software company in Minneapolis bikes to work during the summer months. He can travel to work using one of three routes and wonders whether the average commute times (in minutes) differ between the three routes. He obtains the following data after traveling each route for one week.

Use Tukey’s HSD method at the 5% significance level to determine which routes' average times differ. (Round difference to 1 decimal place, confidence interval bounds to 2 decimal places, and p-values to 3.)

You are designing a new system for detecting an explosive device. Your current design has a 1% false negative rate and a 5% false positive rate. The estimated frequency of actual devices through a typical security check point where the system will be employed is 1 in 5,000. Use the variables E to represent the explosive device and D (or D1 and D2) to represent the detector(s).

a)What is the probability that when the detector indicates the presence of an explosive device that the person is actually carrying one? Show your work?

b)On average, how many people will have to be detained and inspected for every device detected?

c)Now suppose you have additional equipment that gives you an independent test for the same types of devices, but it’s noisier: the false negative rate is 5% and the false positive rate is 10%. What is the probability of a device when both of them signal a detection? Show your work.

4. Suppose from our class (with 25 students), we find the average GPA is 2.7, with standard deviation
0.4. Regard our class as a random sample from the whole university. Based on the information from our
class, can we believe at significance level 0.05, that the average GPA for all students can be at least 2.8?

Independent Sample T Test (Student Height)

Open College Student Data

Research Question: Is there a significant difference between genders on average student height?

Record the following:

1)What test will you run to answer this research question? Why?

2)Is Assumption One: Equal Variances met? How do you know?

3)Is Assumption Two: Normality met? How do you know?

4)Is Assumption Three: Independency met? How do you know?

5)What are the means? (Male and Female)

6)Answer the research question. How do you know?

7)Which gender is statistically taller? How do you know?

8)Is there an effect size (Cohen’s D)? If so what is it? How did you arrive at the effect size? Is it small, medium, large or very large? How do you know?

DATA SET

height   gender (1 - male, 2 - female)
67.00   2
72.00   1
61.00   2
71.00   1
65.00   2
67.00   1
69.00   1
75.00   1
62.00   2
61.00   2
64.00   2
64.00   1
70.00   1
63.00   2
64.00   2
63.00   2
65.00   2
71.00   1
72.00   1
68.00   2
75.00   1
67.00   2
69.00   1
67.00   1
64.00   2
64.00   2
70.00   1
64.00   1
70.00   1
72.00   1
64.00   2
71.00   1
67.00   2
63.00   2
69.00   1
68.00   1
64.00   2
70.00   1
71.00   1
72.00   1
60.00   2
65.00   2
72.00   1
63.00   2
75.00   1
71.00   1
65.00   2
69.00   1
63.00   2
67.00   2

Researchers conducted a study to investigate the effectiveness of two different treatments for depression (Lithium or Imipramine). A sample of individuals diagnosed with bipolar depression were divided into three groups - one group received Lithium, one group received Imipramine, and the last group was given a placebo. After a specified length of time, patients are evaluated for a recurrence of depression. Use StatCrunch to conduct a chi-square test to determine if recurrence is related to the treatment prescribed. In StatCrunch, select Stat > Tables > Contingency > With Data. Select one of the variables of interest for the row variable and the second variable as the column variable. Then click Compute.

• Select the appropriate hypotheses:
  • HoHo: recurrence of depression is related to the treatment prescribed
    HaHa: recurrence of depression is not related to the treatment prescribed
  • HoHo: recurrence of depression is not related to the treatment prescribed
    HaHa: recurrence of depression is related to the treatment prescribed
• TS: χ2χ2 =   (round to 4 decimal places)
• probability =    (round to 4 decimal places)
• Conclusion: At the 0.10 level, there Select an answer is is not  significant evidence to conclude recurrence of depression Select an answer is not is  related to the treatment prescribed.

Throwback Question: Suppose we know that the risk of liver cancer among alcoholics without cirrhosis of the liver is 29.8%. A researcher conjectures that the risk of cancer among alcoholics with cirrhosis of the liver is higher. Suppose we sample 81 alcoholics with cirrhosis of the liver and determine 29 have liver cancer. Use this information to answer the following:

1. For each person in the sample, what variable is recorded?
   • number of years diagnosed with cirrhosis
   • level of alcoholism
   • percentage of individuals with liver cancer
   • if the individual has liver cancer
2. Completely describe the sampling distribution for the sample proportion of alcoholics with cirrhosis who have liver cancer when samples of size 81 are selected (assuming there is no difference in occurrence between those with and without cirrhosis).
   • mean: μˆpμp^  =
   • standard deviation: σˆpσp^ =  (round to 4 decimal places)
   • shape: the distribution of the sample proportion is Select an answer not normally distributed normally distributed   
     • Checks: np =  and n(1 - p) =
3. Test the appropriate claim using the data collected at a significance level of 0.10.
   • HoHo : p ? < > = ≠     HaHa : p ? = < > ≠  
   • αα =
   • TS: z =   (round your final answer to 2 decimal places; carry 4 decimal places throughout your calculation)
   • probability =   (round to four decimal places)
   • Decision: Select an answer fail to reject H₀ reject H₀
   • Interpretation:At the 0.10 level, there Select an answer is is not   significant evidence to support the claim that the incidence rate of liver cancer is Select an answer lower higher different than   in alcoholics diagnosed with cirrhosis.

Regents of a large state university proposed a plan to increase student fees in order to build new parking facilities. A news channel claims that over 70% of the students are opposed to the plan. We wish to test this claim. A random sample of 18 students is taken and 17 of them are opposed to the plan.

A- state the null and alternative hypothesis?

B- estimate the population proportions of students that are opposed to the plan?

C- finding the corresponding standard error for the estimate in part to B and use this standard error to provide an interval estimate for the population proportion with 95% confidence?

D-Calculate the test stastic and provide the p-value for testing the hypothesis. show all work?

E- give a one sentence conclusion in the contexet of this problem?

1. Suppose a random sample of 100 Utah families yields a total fertility rate (the average number of children born per woman) of 2.71. Given what the Central Limit Theorem tells us about the relationship between sample statistics and population parameters, we can assume the total fertility rate for all Utah families is

Exactly 2.71. No more, no less.

Less than 2.71, as the sampling distribution for the TFR is always negatively skewed

likely close to 2.71, though not necessarily EXACTLY 2.71

Far too high relative to the number of students our public education system can realistically support (this is NOT the correct answer)

2. Suppose you're interested in tracking Americans' church attendance. A sample of 900 Americans finds that 22 percent of respondents report attending church at least once per week. If this poll has a margin of error of plus/minus three percent, we can be fairly certain that the percentage of all Americans who attend church at least once per week falls between _______ percent and _______ percent.

Group of answer choices

19; 25

21.7; 22.3

16; 28

There's simply no way of knowing because everyone lies about how often they go to church (this is not the correct answer, although it's probably true).

3. Suppose weights of high school wrestlers are normally distributed with a mean of 145 pounds and a standard deviation of 15 pounds. Approximately what percent of wrestlers weigh between 135 and 150 pounds?

Group of answer choices

37.81 percent

74.75 percent

25.25 percent

0.3781 percent

4. Suppose weights of high school wrestlers are normally distributed with a mean of 145 pounds and a standard deviation of 15 pounds. What weight (approximately) corresponds with the 75^th percentile?

Group of answer choices

134.883

140.22

149.78

155.117

7. A major airline keeps track of data on how their passengers redeem frequent flyer miles. They found that in the last year 58% of passengers redeemed them to purchase tickets for domestic travel, 44% redeemed them to purchase tickets for international travel, and that 16% redeemed them to purchase tickets for both domestic and international travel.

a. What is the probability that in the last year a passenger redeemed frequent flyer miles to purchase a ticket for domestic or international travel?

b. What is the probability that in the last year a passenger did not redeem frequent flyer miles to purchase a ticket for domestic or international travel?

c. Is redeeming frequent flyer miles to purchase a ticket for domestic and international travel mutually exclusive? Explain.

The processing time for the shipping of packages for a company, during the holidays, were recorded for 48 different orders. The mean of the 48 orders is 10.5 days and the standard deviation is 3.08 days. Raw data is given below. Use a 0.05 significance level to test the claim that the mean package processing time is less than 12.0 days. Is the company justified in stating that package processing is completed in under 12 days?

1) Write Ho (null) and H1 (alternative) and indicate which is being tested

2) Perform the statistical test and state your findings; Write answer as a statement