Let us continue with analyses of data from the study of M Daviglus et al (N Engl J Med 1997) on the relationship of fish consumption with death from coronary heart disease (CHD) among 1,822 male employees of the Western Electric Company Hawthorne Works in Chicago who were followed for 30 years. we consider the 242 men who reported consumption of ≥ 35 g/day of fish at baseline. Among these 242 men, 46 died of CHD during follow-up.

a. Provide a point estimate and an appropriate 95% confidence interval for the 30- year risk (probability) of CHD death among employees who typically consume at least 35 g/day of fish. Additionally provide a brief interpretation of your confidence interval.

b. Perform a one-sample test to determine whether the 30-year risk (probability) of CHD death in those who consumed at least 35 g/day of fish is significantly different from 0.25. Make sure to specify your null and alternative hypotheses and give a brief conclusion.

Suppose a die is rolled six times and you need to find

a) The probability that at least two 4 come up

b) The probability that at least five 4's come up

Solve using the Binomial probability formula.

Assume that female students’ heights are normally distributed with a mean given by µ = 64.2 in. and a standard deviation given by σ =2.6 in.

a) If one female student is randomly selected, find the probability that her height is between 64.2 inches and 66.2 inches

b) If 25 female students are randomly selected, find the probability that they have a mean height between 64.2 inches and 66.2 inches

1.Why/When would you use a screening design?

2. What does an orthogonal design mean?

3.T or F: A 2-factor design is said to be balanced if each factor is run an unequal number of times at the high and low levels.

M12 Q21

Professor Gill has taught General Psychology for many years. During the semester, she gives three multiple-choice exams, each worth 100 points. At the end of the course, Dr. Gill gives a comprehensive final worth 200 points. Let x₁, x₂, and x₃ represent a student's scores on exams 1, 2, and 3, respectively. Let x₄ represent the student's score on the final exam. Last semester Dr. Gill had 25 students in her class. The student exam scores are shown below.

Since Professor Gill has not changed the course much from last semester to the present semester, the preceding data should be useful for constructing a regression model that describes this semester as well.

(a) Generate summary statistics, including the mean and standard deviation of each variable. Compute the coefficient of variation for each variable. (Use 2 decimal places.)

Relative to its mean, would you say that each exam had about the same spread of scores? Most professors do not wish to give an exam that is extremely easy or extremely hard. Would you say that all of the exams were about the same level of difficulty? (Consider both means and spread of test scores.)

No, the spread is different; Yes, the tests are about the same level of difficulty.

Yes, the spread is about the same; Yes, the tests are about the same level of difficulty.     

No, the spread is different; No, the tests have different levels of difficulty.

Yes, the spread is about the same; No, the tests have different levels of difficulty.

(b) For each pair of variables, generate the correlation coefficient r. Compute the corresponding coefficient of determination r². (Use 3 decimal places.)

Of the three exams 1, 2, and 3, which do you think had the most influence on the final exam 4? Although one exam had more influence on the final exam, did the other two exams still have a lot of influence on the final? Explain each answer.

Exam 3 because it has the highest correlation with Exam 4; No, the other 2 exams do not have a lot of influence because of their low correlations with exam 4.

Exam 2 because it has the lowest correlation with Exam 4; Yes, the other 2 exams still have a lot of influence because of their high correlations with exam 4.     

Exam 3 because it has the highest correlation with Exam 4; Yes, the other 2 exams still have a lot of influence because of their high correlations with exam 4.

Exam 1 because it has the highest correlation with Exam 4; Yes, the other 2 exams still have a lot of influence because of their high correlations with exam 4.

(c) Perform a regression analysis with x₄ as the response variable. Use x₁, x₂, and x₃ as explanatory variables. Look at the coefficient of multiple determination. What percentage of the variation in x₄ can be explained by the corresponding variations in x₁, x₂, and x₃ taken together? (Use 1 decimal place.)
%

(d) Write out the regression equation. (Use 2 decimal places.)

Explain how each coefficient can be thought of as a slope.

If we hold all other explanatory variables as fixed constants, then we can look at one coefficient as a "slope."

If we hold all explanatory variables as fixed constants, the intercept can be thought of as a "slope."     

If we look at all coefficients together, the sum of them can be thought of as the overall "slope" of the regression line.

If we look at all coefficients together, each one can be thought of as a "slope."

If a student were to study "extra hard" for exam 3 and increase his or her score on that exam by 13 points, what corresponding change would you expect on the final exam? (Assume that exams 1 and 2 remain "fixed" in their scores.) (Use 1 decimal place.)


(e) Test each coefficient in the regression equation to determine if it is zero or not zero. Use level of significance 5%. (Use 2 decimal places for t and 3 decimal places for the P-value.)

Conclusion

We reject all null hypotheses, there is insufficient evidence that β₁, β₂ and β₃ differ from 0.

We reject all null hypotheses, there is sufficient evidence that β₁, β₂ and β₃ differ from 0.     

We fail to reject all null hypotheses, there is sufficient evidence that β₁, β₂ and β₃ differ from 0.

We fail to reject all null hypotheses, there is insufficient evidence that β₁, β₂ and β₃ differ from 0.

Why would the outcome of each hypothesis test help us decide whether or not a given variable should be used in the regression equation?

If a coefficient is found to be not different from 0, then it contributes to the regression equation.

If a coefficient is found to be different from 0, then it does not contribute to the regression equation.     

If a coefficient is found to be not different from 0, then it does not contribute to the regression equation.

If a coefficient is found to be different from 0, then it contributes to the regression equation.

(f) Find a 90% confidence interval for each coefficient. (Use 2 decimal places.)

(g) This semester Susan has scores of 68, 72, and 75 on exams 1, 2, and 3, respectively. Make a prediction for Susan's score on the final exam and find a 90% confidence interval for your prediction (if your software supports prediction intervals). (Round all answers to nearest integer.)

a) An environmental conservation agency recently claimed that more than 30% of Canadian consumers have stopped buying a certain product because the manufacturing of the product pollutes the environment. You want to test this claim. To do so, you randomly select 980 Canadian consumers and find that 314 have stopped buying this product because of pollution concerns. At a = 0.05, can you support the agency’s claim?

*please round your p-hat to 4 decimals before substituting in the z-statistic formula*

b) Refer to question (b). Construct a confidence interval for the true proportion of Canadian consumers who have stopped buying the product at the following levels of confidence: i). 90% ii). 95%

part 1.
An independent measures study has df = 48. How many total participants were in the study?
a. 24
b. 46
c. 50
d. There is not enough information

part 2.

A commonly cited standard for one-way length (duration) of school bus rides for elementary school children is 30 minutes.

A local government office in a rural area conducts a study to determine if elementary schoolers in their district have a longer average one-way commute time. If they determine that the average commute time of students in their district is significantly higher than the commonly cited standard they will invest in increasing the number of school busses to help shorten commute time. What would a Type 2 error mean in this context?

a. The local government decides that the average commute time is 30 minutes.

b. The local government decides that the data provide convincing evidence of an average commute time higher than 30 minutes, when the true average commute time is in fact 30 minutes.

c. The local government decides that the data do not provide convincing evidence of an average commute time higher than 30 minutes, when the true average commute time is in fact higher than 30 minutes.

d. The local government decides that the data do not provide convincing evidence of an average commute time different than 30 minutes, when the true average commute time is in fact 30 minutes.

Consider the following hypothesis test: H0: μ = 15 Ha: μ ≠ 15 A sample of 50 provided a sample mean of 14.12. The population standard deviation is 4. a. Compute the value of the test statistic (to 2 decimals). b. What is the p-value (to 4 decimals)? c. Using α = .05, can it be concluded that the population mean is not equal to 15? Answer the next three questions using the critical value approach. d. Using α = .05, what are the critical values for the test statistic? (+ or -) e. State the rejection rule: Reject H0 if z is the lower critical value and is the upper critical value. f. Can it be concluded that the population mean is not equal to 15?

A school psychologist believes that more positive mood is associated with more creativity. Below are the data from a random sample of 4th graders. What can be concluded with α = 0.05?

a) What is the appropriate statistic?
---Select--- na Correlation Slope Chi-Square
Compute the statistic selected above:  

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 =  ; test statistic =  
Decision:  ---Select--- Reject H0 Fail to reject H0

c) Compute the corresponding effect size(s) and indicate magnitude(s).
If not appropriate, input and/or select "na" below.
effect size =  ;   ---Select--- na trivial effect small effect medium effect large effect

d) Make an interpretation based on the results.

There is a significant positive relationship between positive mood and creativity.There is a significant negative relationship between positive mood and creativity.    There is no significant relationship between positive mood and creativity.

A neighborhood council is interested in the family income and medical care expenditures of its community. In particular, it is believed that lower income is related to more to medical care expenditures. Below are family income (per 1,000 dollars) and medical care expenditure (per 100 dollars) data from a random sample of households in the community. What can be concluded with an α of 0.05?

a) What is the appropriate statistic?
---Select--- na Correlation Slope Chi-Square
Compute the statistic selected above:  

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 =  ; test statistic =  
Decision:  ---Select--- Reject H0 Fail to reject H0

c) Compute the corresponding effect size(s) and indicate magnitude(s).
If not appropriate, input and/or select "na" below.
effect size =  ;   ---Select--- na trivial effect small effect medium effect large effect

d) Make an interpretation based on the results.

There was a significant positive relationship between family income and medical care expenditures.There was a significant negative relationship between family income and medical care expenditures.    There was no significant relationship between family income and medical care expenditures.

Answer the correlation questions using the data below. Use α = 0.05.

a) Compute the correlation.
r =  

b) Compute the appropriate test statistic(s) for H₁: ρ > 0.
critical value =  ; test statistic =  
Decision:  ---Select--- Reject H0 Fail to reject H0

c) Compute the corresponding effect size(s) and indicate magnitude(s).
If not appropriate, input and/or select "na" below.
effect size =  ;  ---Select--- na trivial effect small effect medium effect large effect

d) Make an interpretation based on the results.

There is a significant positive relationship between x and y.There is a significant negative relationship between x and y.    There is no significant relationship between x and y.

A few years​ ago, a certain company introduced a line of​ new, slick swimsuits. Some say that they gave the wearers an advantage in races. In order to test whether the suits were​ effective, suppose that there are 80 swimmers;40 of them are​ professional-level swimmers, and 40 are​ amateur-level swimmers. The designers will ask the swimmers to swim 200metres as fast as possible. It is reasonable to assume that the effects of the suits​ (due to dynamic forces of the​ water) might be different for the two levels of swimmers.

Describe a simple randomized design​ (not blocked) to test whether the slick suits decrease race times. Explain how to assign the swimmers to treatment groups. Choose the correct answer below.

A.Have each simmer wear a slick suit for a 200​-metre race. Record each​ swimmer's time. Ask each swimmer if this time decreased from his or her normal 200​-metre

time.

B.Randomly assign each swimmer to wear either a slick or a​ non-slick suit. Place in a bag 40 tickets that say​ "slick" and 40 that say​ "non-slick." Have each swimmer choose a ticket and use that type of suit in a 200​-metre race. Record each​ swimmer's time.

C.Randomly assign a type of suit to each level of swimmers. Place 2 tickets in a​ bag, one that says​ "slick" and one that says​ "non-slick." Pick one​ ticket, and assign that type of suit to the professional swimmers and the other type of suit to the amateur swimmers. Have them swim a 200​-metre race. Record each​swimmer's time.

D.Let each swimmer choose whether they want to wear a slick suit or a​ non-slick suit, and then have them swim a 200​-metre race. Record each​ swimmer's time.

A frequency distribution is shown below. Complete parts​ (a) through​ (e). The number of dogs per household in a small town.

​(a) Use the frequency distribution to construct a probability distribution. ​(b) Find the mean of the probability distribution. ​(c) Find the variance of the probability distribution.  (d) Find the standard deviation of the probability distribution. ​(e) Using the found values of the mean and the standard​ deviation, an interpretation of the results in the context of the​ real-life situation is that a household on average has _ dog with a standard deviation of _ dog.

Dogs x=0 1 2 3 4 5

Households p(x)= 1225 408 164 44 25 15