A psychologist is interested in the mean IQ score of a given group of children. It is known that the IQ scores of the group have a standard deviation of 11

. The psychologist randomly selects 150 children from this group and finds that their mean IQ score is 99 . Based on this sample, find a 95% confidence interval for the true mean IQ score for all children of this group. Then complete the table below.

Total plasma volume is important in determining the required plasma component in blood replacement therapy for a person undergoing surgery. Plasma volume is influenced by the overall health and physical activity of an individual. Suppose that a random sample of 45 male firefighters are tested and that they have a plasma volume sample mean of x = 37.5 ml/kg (milliliters plasma per kilogram body weight). Assume that σ = 7.30 ml/kg for the distribution of blood plasma.

(a)

Find a 99% confidence interval for the population mean blood plasma volume in male firefighters. What is the margin of error?

(b)

What conditions are necessary for your calculations?

(c)

Interpret your results in the context of this problem.

(d)

Find the sample size necessary for a 99% confidence level with maximal margin of error E = 2.70 for the mean plasma volume in male firefighters.

M3_A5. A political polling organization has been hired to conduct a poll of likely voters prior to an upcoming election. Each voter is to be interviewed in person. It is known that he costs of interviewing different types of voters vary due to the difference in proportion within the population. The costs to interview males, for example, are $20 per Democrat, $18 per Republican, and $27 per Independent voter. The costs to interview female are $24, $22 and $28 for Democrat, Republican, and Independent voters, respectively. The polling service has been given certain criteria to which it must adhere:  There must be at least 7,500 total interviews.  At least 2,500 independent voters must be polled  At least 3,800 males must be polled.  At least 3,250 females must be polled  No more than 35% of those polled may be Democrats  No more than 35% of those polled may be Republicans  No more than 15% of those polled may be Republican males.  Each of the six types of voters must be represented in the poll by at least 13% of the total interviews Formulate and solve this problem in Excel to determine the number of each type of voter that meets the requirements and minimize the cost to carry out the interviews. a) How many decision variables does this problem have? b) Not counting the non-negativity constraint - how many constraints does this problem have? c) What is the minimum cost in your optimal solution (the value of the objective function)? d) Based on your solution – how many Democrat Males should be interviewed? e) Based on your solution – how many Republican Males should be interviewed? f) Based on your solution – how many Independent Males should be interviewed? g) Based on your solution – how many Democrat Females should be interviewed? h) Based on your solution – how many Republican Females should be interviewed? i) Based on your solution – how many Independent Females should be interviewed?

(A) Three marksmen fire simultaneously and independently at a target. What is the probability of the target being hit at least once, given that marksman one hits a target nine times out of ten, marksman two hits a target eight times out of ten while marksman three only hits a target one out of every two times. (B) Fifty teams compete in a student programming competition. It has been observed that 60% of the teams use the programming language C while the others use C++, and experience has shown that teams who program in C are twice as likely to win as those who use C++. Furthermore, ten teams who use C++ include a graduate student, while only four of those who use C include a graduate student. (a) What is the probability that the winning team programs in C? (b) What is the probability that the winning team programs in C and includes a graduate student? (c) What is the probability that the winning team includes a graduate student? (d) Given that the winning team includes a graduate student, what is the probability that team programmed in C? (C) A brand new light bulb is placed in a socket and the time it takes until it burns out is measured. Describe an appropriate sample space for this experiment. Use mathematical set notation to describe the following events: (a) A = thelight bulb lasts at least 100 hours. (b) B = thelight bulb lasts between 120 and 160 hours. (c) C = thelight bulb lasts less than 200 hours. (D) A university professor drives from his home in Cary to his university office in Raleigh each day. His car, which is rather old, fails to start one out of every eight times and he ends up taking his wife’s car. Furthermore, the rate of growth of Cary is so high that traffic problems are common. The professor finds that 70% of the time, traffic is so bad that he is forced to drive fast his preferred exit off the beltline, Western Boulevard, and take the next exit, Hillsborough street. What is the probability of seeing this professor driving to his office along Hillsborough street, in his wife’s car?

a) Explain briefly what this means in practice that the length of the pipes is expected to be µ = 12. What is the probability that a randomly selected pipe is longer than 12.2 meters? What is the probability that the length of a randomly selected pipe is between 11.9 and 12.1 meters? What length is 90% of the tubes longer than? The company has been commissioned to produce a 10.8 kilometer (ie 10800 meters) long pipeline. The has a margin of error of ± 5 meters, that is, they can deliver a pipeline that is up to 5 meters shorter or 5 meters longer than the specified length of 10800 meters. The company is thinking of producing 900 pipes to the pipeline, and the total length of the pipeline then becomes the sum of the lengths of these pipes. Suppose each tube produced by the machine to the company has a length that is independent of the others tubes. b) What is the expected total length of the pipeline? What is the variance of the total length? What is the probability that the pipeline will be too long or too short (ie get a length beyond the allowed margin of error)? If the standard deviation of the lengths of the pipes, σ, could be adjusted to a different value, which value had to be adjusted so that the likelihood of the pipeline becoming too long or too long card to be 0.01? Before production is fully commissioned, one of the engineers at the company insists that they have to examine if the machine is properly adjusted so that the expected length of the pipes is really 12 meters. To examine this, they produce 12 tubes and measure their length. The results of the measurements are given below. We assume in the rest of the task that µ is unknown, while we still have that σ = 0.1 and that the lengths of different pipes are independent. M˚aleresultater: 11.87 11.82 11.99 12.01 11.93 11.98 12.08 12.11 11.92 11.79 12.02 12.07 c) Calculate the mean and median of the data. Find a 95% confidence interval for µ. How many measurements must be made in this situation at least

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

A random sample of 5060 permanent dwellings on an entire reservation showed that 1648 were traditional hogans.

(a) Let p be the proportion of all permanent dwellings on the entire reservation that are traditional hogans. Find a point estimate for p. (Round your answer to four decimal places.)


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

Give a brief interpretation of the confidence interval.

99% of the confidence intervals created using this method would include the true proportion of traditional hogans.

99% of all confidence intervals would include the true proportion of traditional hogans.     

1% of all confidence intervals would include the true proportion of traditional hogans.

1% of the confidence intervals created using this method would include the true proportion of traditional hogans.

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

No, the conditions are not 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.     

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.

Suppose you work for a pharmaceutical company that is testing a new drug to treat ulcerative colitis. You conduct a random sample of 31 patients on the new drug and 10 of them indicated a decrease in symptoms since starting the drug regimen. You construct a 95% confidence interval for this proportion to be ( 0.158 , 0.4871 ). What is the correct interpretation of this confidence interval?

Question 9 options:

The company has been making efforts to increase the skills of its workers and they have been funding training programs for employees. Five years ago, only 8% of the workforce had C TRAINING LEVEL, the most advanced training level.

They will use this sample to determine if there is significant evidence that their efforts have led to an increase in the proportion of employees with C  TRAINING LEVEL. Use significant level alpha=5% to conduct the appropriate hypothesis test and enter your results and conclusion below.

Select a conclusion and enter the corresponding letter of your choice on the answer field :

1. There is evidence that the proportion of employees with C TRAINING LEVEL has increased.
2. There is NO evidence that the proportion of employees with C TRAINING LEVEL has increased.

Instructions: Be sure to define all parameters and random variables used. Also, you must provide any code you use (final computed values are not sufficient). I strongly recommend that you create Jupyter Notebooks for this assignment, embedding your written comments in Markdown cells.

6. A biologist wishes to examine the collision avoidance behaviour of bullfrogs housed in experimental boxes. He placed an object in the upper visual field of each frog and then moved the object progressively closer until the frog jumped to avoid collision. He is interested in how the angle (in degrees) of a frog's escape direction depends on its angular velocity (degrees/second). The data (for only the 69 frogs whose escape direction was less than 50 degrees) are available in the file frog50.csv.

a) Give the equation of the fitted regression line.

b) Create a plot of direction vs. velocity, including the fitted regression line

c) Interpret the slope of the fitted regression line in the language of the problem

d) State the R2 value and its interpretation (in the language of the problem)

e) What is the best prediction of the velocity of a jump with a direction of 25 degrees?

f) Can you provide an accurate prediction of the velocity of a jump with direction of 60 degrees? If so, provide this prediction. If not, explain why

frog50.csv.file

"Direction","Velocity"

1.9,22.3

4.9,34.8

6.6,57.3

16.9,124.6

5.8,14.1

32,191.6

26,129.3

1.1,8.5

6.7,33.1

1.7,1.7

19.1,125.3

9.5,35.9

9.2,50.9

35.2,183.2

7.3,53.2

47.5,286.7

37.5,173.4

4.8,19.8

21.4,90.3

1.4,7.2

1.9,7.5

2.4,10.6

1,17.5

1.3,6.1

11.7,83.8

24.3,193.2

1.4,27

1.6,11.5

9.7,59.2

44.4,265.8

4.6,5.1

4.2,31

1.8,17.3

21.3,154.6

19.1,88.5

4.9,10

15.6,96.6

30,178

38.1,299.6

1.2,7.1

12.5,70.4

43.7,269.8

6.7,9.1

1.1,0.6

32,237.2

8.1,35.9

16.6,74.3

25.1,130.3

2.7,26

1.7,17.4

6.9,39.2

8.4,17.2

2,3.7

47.5,280.2

2.9,0.1

1.2,8.8

15.4,97.4

19.8,104.5

3.8,12.4

16.1,91

1.5,8

6.3,51.7

25,157.1

3,10

36.6,193.8

5.3,21.4

4.5,13.5

3.8,43.4

3.1,27.8

Can any linear regression model be checked for model adequacy by statistical testing for lack of fit or goodness of fit? Why or why not? Please provide your answer with detailed justification (i.e., by mathematical proof or by showing a numerical example)

4. The owner of Bun ‘N’ Run Hamburgers wishes to compare the sales per day at two locations. The mean number sold for 10 randomly selected days at the Northside site was 83.55, and the standard deviation was 10.50. For a random sample of 12 days at the Southside location, the mean number sold was 78.80 and the standard deviation was 14.25. At the .05 significance level, is there a difference in the mean number of hamburgers sold at the two locations? Assume variances are equal. (29)

According to Masterfoods, the company that manufactures M&M’s, 12% of peanut M&M’s are brown, 15% are yellow, 12% are red, 23% are blue, 23% are orange and 15% are green. You randomly select six peanut M&M’s from an extra-large bag of the candies. (Round all probabilities below to four decimal places; i.e. your answer should look like 0.1234, not 0.1234444 or 12.34%.)

Compute the probability that exactly two of the six M&M’s are red. ________

Compute the probability that two or three of the six M&M’s are red. _______

Compute the probability that at most two of the six M&M’s are red. _______

Compute the probability that at least two of the six M&M’s are red. _______

If you repeatedly select random samples of six peanut M&M’s, on average how many do you expect to be red? (Round your answer to two decimal places.) ____red M&M’s

With what standard deviation? ________(Round your answer to two decimal places.) red M&M’s

1. Consider simple regression model (one independent variable)

a) How do you find a confidence interval for the coefficient of X?

b) How do you conduct a test if the coefficient of X is 10 or not?

c) How do you find a CI for the mean of the dependent variable if x=2?

The following is the regression output for a study on serious crime rates in cities in the United States. The data was collected in the 1970s. AREA is the size of the city, DOCS is the number of Doctors, % > 65 is the proportion of residents greater than 65 years of age and HOSPITAL BEDS is the number of hospital beds per population. Answer the following questions given the supplied regression output. These variables are regressed on the serious crime rates in an attempt to explain variance in crime rates in different cities. (3 points per question)

1. How accurate is regression model? What amount of the variance in Serious Crimes is explained
2. How strong is the linear relationship between the predictors and Serious Crimes?
3. How close to the regression line is the majority of observations?
4. Does the model fit the data better than not using these predictor variables? Is the F-test significant? What hypothesis can we draw from it?
5. Use the output to form the regression equation.
6. Are all the predictors significant? Is zero contained in the confidence intervals? Why is that interesting?
7. Of the significant variables, does the corresponding slopes make intuitive sense?   Are the effects of the predictors reasonable?
8. How much variance in serious crime is unaccounted for? What additional data might explain some of this remaining variance?