The following regression output was obtained from a study of architectural firms. The dependent variable is the total amount of fees in millions of dollars.

X1 - # of architects employed by the company

X2 - # of engineers employed by the company

X3 - # of years involved with health care projects

X4 - # of states in which the firm operates

X5 - % of the firms work that is health care-related

1. Write out the regression equation
2. How large is the sample? How many independent variables are there?
3. Conduct a global test of hypothesis to see if any of the set of regression coefficients could be different from 0. Use the .05 significance level. What is your conclusion?
4. Conduct a test of hypothesis for each independent variable. Use the .05 significance level. Which variable would you consider eliminating first?
5. Outline a strategy for deleting independent variables in this cas

Use R to complete the following questions. You should include your R code, output and plots in your answer.

1. Two methods of generating a standard normal random variable are:

a. Take the sum of 5 uniform (0,1) random numbers and scale to have mean 0 and standard deviation 1. (Use the properties of the uniform distribution to determine the required transformation).

b. Generate a standard uniform and then apply inverse cdf function to obtain a normal random variate (Hint: use qnorm).

For each method generate 10,000 random numbers and check the distribution using

a. Normal probability plot

b. Mean and standard deviation

c. The proportion of the data lying within the theoretical 2.5 and 97.5 percentiles and the 0.5 and 99.5 percentiles. (Hint: The ifelse function will be useful)  

If the probability that a family will buy a vacation home in Manmi, Malibu, or Newport is 0.25, 0.10, 0.35, what is the probability the family will consummate one of these transactions? please show all wor with explanation.

In the following problem, check that it is appropriate to use the normal approximation to the binomial. Then use the normal distribution to estimate the requested probabilities.

What are the chances that a person who is murdered actually knew the murderer? The answer to this question explains why a lot of police detective work begins with relatives and friends of the victim! About 68% of people who are murdered actually knew the person who committed the murder.† Suppose that a detective file in New Orleans has 65 current unsolved murders. Find the following probabilities. (Round your answers to four decimal places.)

(a) at least 35 of the victims knew their murderers

(b) at most 48 of the victims knew their murderers

(c) fewer than 30 victims did not know their murderers

(d) more than 20 victims did not know their murderers

Mr. James McWhinney, president of Daniel-James Financial Services, believes there is a relationship between the number of client contacts and the dollar amount of sales. To document this assertion, Mr. McWhinney gathered the following sample information. The X column indicates the number of client contacts last month, and the Y column shows the value of sales ($ thousands) last month for each client sampled.

1. Determine the standard error of estimate
2. Determine the coefficient of determination. Interpret the coefficient of determination

Professor Nord stated that the mean score on the final exam from all the years he has been teaching is a 79%. Colby was in his most recent class, and his class’s mean score on the final exam was 82%. Colby decided to run a hypothesis test to determine if the mean score of his class was significantly greater than the mean score of the population. α = .01.  If p = 0.29

What is the mean score of the population? What is the mean score of the sample? What should Colby’s statement of conclusion be

You started taking the bus to work. The local transit authority says that a bus should arrive at your bus stop every five minutes. After a while, you notice you spend a lot more than five minutes waiting for the bus, so you start to keep a record.

You spend the next two months recording how long it takes for the bus to arrive to the bus stop. This give a total of sixty observations that denote the number of minutes it took for the bus to arrive (rounded to the nearest minute). These observations are hosted at

   https://mattbutner.github.io/data/bus_stop_time.csv

Load these data into R as a data frame titled bus_stop_time

Create a histogram of the time_until_bus varaible. Would you say that five minutes is a reasonable guess for the average arrival time based on this picture alone?

Create 95% confidence interval for the bus arrival times using the Z distribution. Does 5 minutes fall within the 95% confidence interval?

How would you communicate your finding to the local transit authority?

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 63 professional actors, it was found that 39 were extroverts.

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


(b) Find a 95% 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 5% confident that the true proportion of actors who are extroverts falls within this interval.We are 95% confident that the true proportion of actors who are extroverts falls outside this interval.    We are 5% confident that the true proportion of actors who are extroverts falls above this interval.We are 95% confident that the true proportion of actors who are extroverts 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.No, the conditions are not satisfied. This is important because it allows us to say that p̂ is approximately normal.    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.

A study  examined whether or not noses continue to grow throughout a person’s lifetime.  The study included many measurements (including size of the nose as measured by total volume) and included multiple tests.  For each of the tests described below:

            a)  State the null and alternative hypotheses.

            b) Give a formal decision using a 5% significance level, and interpret the conclusion in context.

• In a test to see whether males, on average, have bigger noses than females, the study indicates that “p < 0.01.”
• In a test to see whether there is a positive linear relationship between age and nose size, the study indicates that “p < 0.001.”

The failure rate in a statistics class is 20%. In a class of 30 students, find the probability that exactly five students will fail. Use the normal distribution to approximate the binomial distribution.

The mean number of pets per household is 2.96 with standard deviation 1.4. A sample of 52 households is drawn. Find the probability that the sample mean is less than 3.11.

The pH of 20 randomly selected lakes is measured. Their average pH is 5.7.

Part A. Historically the standard deviation in the pH values is 0.9. Use this standard deviation for the following questions.
Part Ai. Build a 95% confidence interval for the population mean lake pH.
Part Aii. Build a 90% confidence interval for the population mean lake pH.
Part Aiii. Build an 80% confidence interval for the population mean lake pH.
Part Aiv. Compare the intervals you created in Ai, Aii and Aiii. What effect does changing the level of confidence have on the interval?

Please solve only part B

Part B. For the 20 measured lakes the standard deviation in the pH values is 0.9. Use this standard deviation for the following questions.
Part Bi. Build a 95% confidence interval for the population mean lake pH.
Part Bii. Compare the confidence intervals in Bi and Ai. What effect does not knowing the value of ? have on the interval?
Part Biii. Test whether the population mean pH differs from 6.

For 300 trading​ days, the daily closing price of a stock​ (in $) is well modeled by a Normal model with a mean of

​$197.12197.12 and a standard deviation of

​$7.187.18. According to this​ model, what is the probability that on a randomly selected day in this​ period, the stock price closed as follows.

​a) above ​$204.30204.30​?

​b) below ​$211.48211.48​?

​c) between ​$182.76182.76 and ​$211.48211.48​?