A student is taking a quiz with nine questions. Each question has answers: A, B, C, or D. The student guesses on each question
     
      a) how many repetitions for this problem?
      b) what is the probability of success on the 1st repetition?
   c) what is the probability of guessing 3 answers correctly?

d) what is the probability of guessing at most 2 correct?

Business Weekly conducted a survey of graduates from 30 top MBA programs. On the basis of the survey, assume the mean annual salary for graduates 10 years after graduation is $121,000. Assume the standard deviation is $42,000. Suppose you take a simple random sample of 42 graduates. Round all answers to four decimal places if necessary. What is the distribution of X ? X ~ N( , ) What is the distribution of ¯ x ? ¯ x ~ N( , ) For a single randomly selected graduate, find the probability that her salary is between $122,679 and $130,519. For a simple random sample of 42 graduates, find the probability that the average salary is between $122,679 and $130,519. For part d), is the assumption of normal necessary? YesNo

Problem 4. The data {(1, 10),(2, 5.49),(3, 0.89),(4, −0.14),(5, −1.07),(6, 0.84)} comes from a model F(x) = (r/ x )+ sx. Use least squares to estimate the parameters r, s.

The manager of a computer retails store is concerned that his suppliers have been giving him laptop computers with lower than average quality. His research shows that replacement times for the model laptop of concern are normally distributed with a mean of 3.7 years and a standard deviation of 0.5 years. He then randomly selects records on 37 laptops sold in the past. Round the answers of following questions to 4 decimal places. What is the distribution of X ? X ~ N( , ) What is the distribution of ¯ x ? ¯ x ~ N( , ) What is the probability that one randomly selected laptop is replaced less than 3.6 years? For 37 laptops, find the probability that the average replacement time is less than 3.6 year. For part d), is the assumption of normal necessary? YesNo

In a survey of 2508 2508 adults in a recent​ year, 1395 1395 say they have made a New​ Year's resolution. Construct​ 90% and​ 95% confidence intervals for the population proportion. Interpret the results and compare the widths of the confidence intervals. The​ 90% confidence interval for the population proportion p is left parenthesis nothing comma nothing right parenthesis , . ​(Round to three decimal places as​ needed.) The​ 95% confidence interval for the population proportion p is left parenthesis nothing comma nothing right parenthesis , . ​(Round to three decimal places as​ needed.) With the given​ confidence, it can be said that the ▼ of adults who say they have made a New​ Year's resolution is ▼ between the endpoints less than the upper endpoint not between the endpoints greater than the lower endpoint of the given confidence interval. Compare the widths of the confidence intervals. Choose the correct answer below. A. The​ 90% confidence interval is wider. B. The​ 95% confidence interval is wider. C. The confidence intervals cannot be compared. D. The confidence intervals are the same width.

Hypotheses are defined as:

• “Hypotheses are single tentative guesses, good hunches—assumed for use in devising theory or planning experiments intended to be given a direct experimental test when possible” (Rogers, 1966).
• “Hypothesis is a formal statement that presents the expected relationship between an independent and dependent variable” (Cresswell, 1994).
• “Hypothesis is a statement or explanation that is suggested by knowledge or observation but has not, yet, been proved or disproved” (Macleod & Hockey, 1981).

The null hypothesis represents a theory that has been put forward, either because it is believed to be true or because it is to be used as a basis for argument but has not been proved. The null hypothesis offers close to null prediction.

The hypothesis actually to be tested is usually given the symbol H0 and is commonly referred to as the null hypothesis. The null hypothesis is assumed to be true unless there is strong evidence to the contrary —similar to how a person is assumed to be innocent until proven guilty.

The alternative hypothesis is a statement of what a hypothesis test is set up to establish. The other hypothesis, which is assumed to be true when the null hypothesis is false, is referred to as the alternative hypothesis and is often symbolized by HA or H1.

• It is opposite the Null Hypothesis.
• It is only reached if H0 is rejected.
• Frequently, the “alternative” is actually the desired conclusion of the researcher!

For this assignment, select two different business research scenarios you are interested in testing.

Write a set of hypotheses for each scenario.

The rental car agency has 30 cars on the lot. 10 are in great shape, 16 are in good shape, and 4 are in poor shape. Four cars are selected at random to be inspected. Do not simplify your answers. Leave in combinatorics form. What is the probability that:

a. Every car selected is in poor shape

b. At least two cars selected are in good shape.

c. Exactly three cars selected are in great shape.

d. Two cars selected are in great shape and two are in good shape.

e. One car selected is in good shape but the other 3 selected are in poor shape.

Bulldog, Inc. is a Texas-based audio equipment manufacturer. It produces two kinds of audio speakers. The first kind is a bookshelf speaker, called BSS. The other kind is a floor speaker, called FLS.

Both BSS and FLS use cherry wood for speaker body structure. Each BSS uses 2 sheets of cherry wood and each FLS uses 5 sheets of cherry wood for production. Each month Bulldog, Inc. has 600 sheets of cherry wood available for production.

According to its metropolitan user survey, most people prefer BSS due to its compact size. Bulldog, Inc. wants BSS’s monthly production to be at least 120 units. Since Bulldog, Inc. only has one warehouse, its total combined production of BSS and FLS cannot exceed 255 units. Currently each BSS can bring in $150 as unit profit and each FLS can bring in $300 as unit profit. Bulldog, Inc. wants to find a production mix for BSS and FLS to maximize Bulldog, Inc. monthly profit.

1) Formulate this problem as a linear program.

2) Solve this linear program graphically for the optimal solution. Report the optimal solution.

3) What is the profit generated by using the above optimal solution?

After you finish the above problem, you received a memo from the Bulldog, Inc. sales manager. According to the ongoing promotion event, the revised unit profit is now $100 for BSS and $400 for FLS based on the new marketing information.

4) What will be your modified optimal solution based on the information from the sales manager?

5) What is the modified profit?

Suppose you gather a sample size of 20 widgets and construct a 99% confidence interval for the mean number of widgets.

Assuming this research follows approximately normal with unknown mean and variance, what would be the value of the distribution percentage point that I would use when constructing this interval?

Thanks for answering!

1) A normal random variable x has an unknown mean μ and standard deviation σ = 2. If the probability that x exceeds 4.6 is 0.8023, find μ. (Round your answer to one decimal place.)

μ =

2) Answer the question for a normal random variable x with mean μ and standard deviation σ specified below. (Round your answer to four decimal places.)

μ = 1.3 and σ = 0.19.

Find

P(1.50 < x < 1.71).

P(1.50 < x < 1.71) =

3) Answer the question for a normal random variable x with mean μ and standard deviation σ specified below. (Round your answer to four decimal places.)

μ = 1.2 and σ = 0.18.

Find

P(x > 1.34).

P(x > 1.34)

4) Answer the question for a normal random variable x with mean μ and standard deviation σ specified below. (Round your answer to four decimal places.)

μ = 1.3 and σ = 0.19.

Find

P(1.00 < x < 1.10).

P(1.00 < x < 1.10)

5) Let z be a standard normal random variable with mean μ = 0 and standard deviation σ = 1. Find the value c that satisfies the inequality. (Round your answer to two decimal places.)

P(−c < z < c) = 0.80

c =

6) Suppose x has a uniform distribution on the interval from −1 to 1. Find the probability.

P(x > 0.5)

Describe a confidence interval for the mean of a population by stating
1. a population and a quantitative variable on that population,
2. a sample size,
3. a sample mean of that variable, and
4. a sample standard deviation, and
5. a confidence level, then
6. finding the interval. Then perform a test of significance on the mean of the population by stating
7. both a null and an alternative hypothesis and
8. an α-level, then finding
9. the one-sample t-statistic and either
10. rejecting or failing to reject the null hypothesis. Remember that you do not need to list the values of the variable for individuals in either the sample or the population, and that the
values for 2, 3, 4, 5, and 8 do not need to be calculated, only stated.

Please provide a simple and easy example with data.

Problem 1 Let X1, X2, . . . , Xn be independent Uniform(0,1) random variables. (

a) Compute the cdf of Y := min(X1, . . . , Xn).

(b) Use (a) to compute the pdf of Y .

(c) Find E(Y ).

Demonstrate understanding of the Central Limit Theorem, using R, by showing how the distribution of the sample mean changes according to sample size.

Consider a Poisson distribution with λ = 1.5.

Generate samples of 10,000 means over different numbers of observations (eg give a matrix 1, 2,3...100) rows. For each of these samples of means, compute the mean of the means, the sample standard deviation of the means, and the proportions of means that are more than 1 standard deviation above the overall mean.

Generate plots of each of these quantities vs the number of observations contributing to the means (1, 2, 3, 4 etc.).

Write R code used to produce these data. Form conclusions about what is seen, based on the Central Limit Theorem.

**Note: need help with the R code that produces this. Please supply that in your answer. Thanks!

1. In the R programming language, we would like to use the data set called iris to build a simple linear regression model to predict

Sepal.Length based on Petal.Length.

1. Calculate the least squares regression line to predict Sepal.Length based on Petal.Length. Interpret the slope of the line in the context of the problem. Remember that both variables are measured in centimeters.
2. Plot the regression line in a scatterplot of Sepal.Length vs. Petal.Length.
3. Test H1: ??₁ ≠ 0 at ?? = 0.05 using both a ??-test and an ??-test. Report the test statistics and interpret the results.
4. Visually check the normality assumption. Does it seem reasonable here?
5. Visually check the constant variance assumption. Does it seem reasonable here?
6. Interpret this regression model’s ??².
7. Using the regression line, what would you predict as the Sepal.Length for an iris with a Petal.Length of 3.4 cm?
8. We would expect approximately 95% of the irises to have a Sepal.Length within ± (fill in the blank)  cm of their predicted values from the regression line.

Answers should be in the form of R code on how to accomplish each part and include the correct statistical explanation for those that require it in the question. Please be as thorough as possible. Thank you so much!!!