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Homework answers / question archive / Question 1 The alternative hypothesis for the one-way analysis of variance states that: 0 A
Question 1
The alternative hypothesis for the one-way analysis of variance states that:
0 A. at least one mean is different from zero
0 B. only one mean is different from all of the other means
0 C. all mean are the same
0 D. at least one of the means is different from all of the other means
0 E. the largest mean is different from the average of the other means
0 F. the largest mean is different from the control mean
0 G. the smallest mean is different from the control mean
0 H. none of the above
Question 2
Consider a fitted regression equation y = 13.7x - 2.3 where y is measured in metres. The fitted equation is based on n observations where the first observation is (x,y) = (4.1, 50.1). What is the residual corresponding to the first observation?
0 A. cannot answer without knowing the sample size n
0 B. cannot answer without knowing the sample means for x and y
0 C. both A and B
0 D. 3.77 metres
0 E. -3.77 metres
0 F. 3.77 0
G. -3.77 0
H. none of the above
Question 3
In the simple linear regression model Y = β0 + β1 X + e, it is standard procedure to plot the residuals against the covariates (predictor variable) X. Indicate which of the following is the only true statement concerning the residual plot.
0 A. the residuals are a surrogate for the unknown β0
0 B. the residuals are a surrogate for the unknown β1,
0 C. the plot tells us about the constancy of variance concerning the unobserved error terms E
0 D. the plot tells us whether we have a sufficiently large dataset
0 E. a pattern of increasing residuals versus X is desirable
0 F. two of A-E are correct
0 G. three of A-E are correct
H. four of A-E are correct
Question 4
Investigators gave caffeine to fruit flies to see if it affected their rest. The four treatments were a control, a low caffeine dose of 1 mg/ml of blood, a medium dose of 3 mg/ml of blood, and a higher caffeine dose of 5 mg/ml of blood. Twelve fruit flies were assigned at random to the four treatments, three to each treatment, and the minutes of rest measured over a 24-hour period were recorded. The data follow.
|
Treatment |
Minutes of Rest |
Minutes of Rest |
Minutes of Rest |
|
Control |
450 |
413 |
418 |
|
Low dose |
466 |
422 |
435 |
|
Medium dose |
421 |
453 |
419 |
|
High dose |
364 |
332 |
389 |
Assume the data are four independent SRSs, one from each of the four populations of caffeine levels, and that the distribution of the yields is Normal. A partial ANOVA table follows, along with the means and standard deviation of the yields for the four levels.
Question 5
In simple linear regression, we have the model Y = β0 + β1 X ∈ and the fitted model ?= β0 + β1 X . Indicate which of the following is the only true statement.
0 A. Y consists of observed data
0 B. β0 consists of observed data
0 C. ∈ consists of observed data
0 D. ? consists of observed data
0 E. β0 consists of observed data
0 F. two of A-E consist of observed data
0 G. three of A-E consist of observed data
0 H. four of A-E consist of observed data
One-way ANOVA: Rest Versus Caffeine
Source DF SS MS F P
Caffeine 11976
Error 588.75
|
Level |
N |
Mean |
St Dev |
|
Control |
3 |
427.00 |
20.07 |
|
Low |
3 |
441.00 |
22.61 |
|
Medium |
3 |
431.00 |
19.08 |
|
High |
3 |
361.00 |
29.61 |
The value of the ANOVA F statistic for testing equality of the population means of the rest times at each level is:
0 A.0.045
0 B.2.78
0 C 4.73
0 D.4.82
0 E. 5.56
0 F. 7.41
0 G. 22.23
0 H. none of the above
A marketing researcher was studying the effect of a supermarket display on sales of a new product. There were two designs for the display. The first had greater visual appeal, and the second contained more factual information about the product. Each type of display could be made in three sizes - small, medium, or large. Eighteen supermarkets were available for the study, and three supermarkets were selected at random for each combination of display design and size. The number of units of the product sold over a two-week period was recorded for each supermarket. For the resulting data, a two-way ANOVA was run with the partial ANOVA table given below.
Analysis of variance for SALES:
|
Source |
DF |
SS |
MS |
P |
|
Design |
|
|
|
0.163 |
|
Size |
|
|
7256 |
|
|
Size*Design |
|
41707 |
|
|
|
Error |
|
|
12262 |
|
|
Total |
17 |
206664 |
|
|
In the ANOVA table, the test for the main effect of Design has a p-value of 0.613. This indicates that,
0 A. sales vary considerably for the different designs
0 B. for about 61.3% of the samples, there was a difference in the effect of design
0 C. for about 61.3% of the samples, there was no difference in the effect of design
0 D. visual appeal has a greater impact than factual information on sales
0 E. factual information has a greater impact than visual appeal on sales
0 F. the reported p-value must be studied in context with the p-value for the effect of size
0 G. there is a 61.3% difference in the effects due to the two designs
0 H. None of the above
Question 7
Sarah performs an experiment to determine the best amount of sun and water for a certain type of plant. She varies the amount of water the plants get each day and also how much time they are in the sun. At the end of six weeks, she measures the height of each plant. In an experiment in which there are two independent variables, how many effects will an ANOVA test for?
0 A. Two main effects
0 B. Two interaction effects
0 C. Two main effects and one interaction effect
0 D. One main effect and one interaction effect
0 E. One main effect
0 F. Two main effects and two interaction effects
0 G. One interaction effect
0 H. None of the above
Question 8
Based on a sample of the salaries of professors at a major university, you have performed a multiple regression relating salary to years of service and gender. The estimated multiple linear regression model is:
Salary = $45,000 + $3000(Years) + $4000(Gender) + $1000(Years)(Gender)
where Gender = 1 if the professor is male and Gender = 0 if the professor is female. Indicate which of the following is the only statement concerning the interaction term.
0 A. there is no interaction term in this regression model
0 B. there is a gap between male and female salaries that widens with years of service
0 C. the interaction term is the intercept
0 D. if gender is coded to take on values 1.2.3. Then it is impossible to assess interaction
0 E. interaction effects only exist with continuous explanatory variables
0 F. none of A-E are correct
0 G. two of A-E are correct
0 H. three of A-E are correct
Question 9
Suppose that (15.6, 18.9) is a 95% confidence interval. Indicate which of the following statement is the only true statement.
Question 10
Indicate which of the following is the only true statement concerning the assumptions of the one-way ANOVA?
A. all the response variables follow the same normal distribution
B. all response variables have the same variance
C. all responses within a group are independent
D. all responses (even those across treatments) are independent
E. subjects must be randomly assigned to the groups
F. two of A-E are correct
G. three of A-E are correct
H. four of A-E are correct
Question 11
Researchers wanted to know whether drawing a happy face on the back of restaurant customers' checks lead to higher tips. Four servers each recorded the tips for 50 consecutive tables. For 25 of the 50 tables, following a pre-determined randomization, they drew a happy face on the back of the check. The other 25 tables received no happy face. The response was the tip expressed as the bill percentage. Indicate which of the following is the only true statement.
o A. This is an experiment rather than an observational study
o B. The degrees of freedom associated with the servers is 4
o C. This is an example of a two-way ANOVA design
o D. If the F statistic for the Happy Face effect is large. it indicates that drawing a happy face affects sales
o E. None of A-D are true
o F. Two of A-D are true
0 G. Three of AD are true
0 H. Four of A-D are true
Question 12
Consider a random variable Y and and an explanatory variable X1. You believe that Y increases as X1 increases, then plateaus, and then Y decreases as X1 increases. Furthermore, you believe that this general pattern holds for two different types of subjects according to X2 = 0 and X2 = 1. Given the physical understanding of the relationship involving Y, X1 and X2, which model is the most appropriate?
o A. Y = β0+ β1 X1+ β2X2 + β3 X1X2 +E
O B.Y = β0+ β1 X1+ β2X2 + β3 X1X2 +E
0 C.Y = β0+ β1 X1+ β2X2 - β3 X1X2 +E
O D.Y = β0+ β1 X2+ β2X2 + β3 X1+ β4 x1x2 + β5x2x1+E
o E.Y = β0+ β1 X1+ β2X2 + β3x1x2+ E
0 F Y = β0+ β1 X1+ β2X2 + β3x1x2+ β4x1x2+ E
0 G Y= β0+ β1 X1+ β2X1 + β3x2+ β4x1x2+β5x1x2+ E E +E
0 H. none of the above
Question 13
In the simple linear regression model Y = β0+ β1 X + €, the most common test concerns the hypothesis H0 : β1 = 0. Indicate which of the following is the only true statement concerning the plot.
o A. if H0 is true, then the response Y does not depend on the predictor X
o B. with observational data, the hypothesis HO investigates the causal relationship between X and Y o C. if H0 is rejected, then β1 provides an estimate of how much Y will change if X is increased by 1 unit
o D. H0 is rejected when β1 >3
0 E. Ho is rejected when β1 >3
F. two of A-E are correct
o G. three of A-E are correct
0 H. four of A-E are correct
Question 14
For fast-food restaurants, many menu items are high in fat, so most of their calorie content comes from fat (rather than carbohydrate or protein). Here we investigate the relationship between the amount of fat in a menu item ("Fat," measured in grams) and the number of calories ("Calories"). To predict the number of calories in a menu item given its fat content, we use the simple linear regression model Calories = a + β (Fat) where the deviations are assumed to be independent and normally distributed, with mean 0 and standard deviation o-.
At one major fast-food restaurant chain, there were 26 items listed under the heading of "Sandwiches" (which includes hamburgers, chicken sandwiches, and other sandwich selections) on the menu. We fit the model described above to the data using the method of least squares. We treat these 26 menu items as a sample from the population of all sandwich items at all fast-food restaurants. The following is a plot of the residuals versus fat for 26 menu items at the fast food restaurant.
Indicate which of the following is the only true statement concerning the plot.
0 A. The estimate of the intercept will be close to zero
0 B. There is evidence that the deviations described by the model are not Normal in distribution
0 C. The outliers and influential observations in the plots means that the assumptions for regression are violated
0 D.A linear model is appropriate for explaining the relationship between the explanatory and response variable
0 E. None of the above are correct
0 F. two of A-E are correct
0 G. three of A-E are correct
0 H. four of A-E are correct
Question 15
For fast-food restaurants, many menu items are high in fat, so most of their calorie content comes from fat (rather than carbohydrate or protein). Here we investigate the relationship between the amount of fat in a menu item ("Fat," measured in grams) and the number of calories ("Calories"). To predict the number of calories in a menu item given its fat content, we use the simple linear regression model Calories = a + 0 (Fat) where the deviations are assumed to be independent and Normally distributed, with mean 0 and standard deviation o-.
At one major fast-food restaurant chain, there were 26 items listed under the heading of "Sandwiches" (which includes hamburgers, chicken sandwiches, and other sandwich selections) on the menu. We fit the model described above to the data using the method of least squares. We treat these 26 menu items as a sample from the population of all sandwich items at all fast-food restaurants. The following results were obtained from software.
r2 =0.846
s = 43.5747
|
Parameter |
Estimate |
Std. Error |
|
Α |
151.092 |
30.082 |
|
Β |
12.546 |
1.21 |
The slope of the least-squares regression line is:
0 A.15109
0 B. 12.546
0 C. 0.846
0 D. 30.082 0
E.1.21 0
F.43.5747
0 G. there is not sufficient information to determine the slope
0 H. none of the above
Question 16
In the simple linear regression model Y= β0 + β1x+e, it is standard procedure to produce a normal quantile plot. Indicate which of the following is the only true statement concerning the plot.
Question 17 1 pts
Consider the regression model Y= β0 + β1x1 + β2X3+ β3 X3 + β4 Xi X3 + β5 X1 X2 + β6 X2 X3 +E. Indicate which of the following is the only true statement concerning nested models.
0 A. Y= β0 + β1x1 + β2X3+ β3 X3+ E is a nested model
0 Y= β0 + β1x1 + β2X3+ β3 X3 + β4 Xi X3 + Eis a nested model
0 Y= β0 + β1x1 + β2X3+ β3 X3 + β4 Xi x2 X3+e is a nested model
0 Y= β0 + β1x1 + β2X3+ β3 X3 + β4 Xi x2 + E is a nested model
O E. none of the models A-D are nested models
O F. exactly two of the models A-D are nested models
O G. exactly three of the models A-D are nested models
0 H. all of the models A-D are nested models
Question 18
The coefficient of multiple determination R2 is a standard diagnostic used in regression. Indicate which of the following is the only true statement concerning the diagnostic.
0 A. it describes the square of the proportion of variability in the explanatory variable that is explained by the response
B. it describes the square of the proportion of variability in the response that is explained by the explanatory variables
0 C. it describes the proportion of variability in the explanatory variable that is explained by the response variables
D. it describes the proportion of variability in the response variable that is explained by the explanatory variables E. the value of the diagnostic lies in the interval (0.1)
F. larger values of the diagnostic are indicative of better fit
G. two of A-F are correct
H. three of A-F are correct
Question 19
In many fast food restaurants. there is a strong correlation between a menu item's fat content (measured in grams) and its calorie content. We want to investigate this relationship. Using all of the food menu items at a well-known fast food restaurant. the fat content and calorie contents were measured. We decide to fit the least-squares regression line to the data. with fat content (x) as the explanatory variable and (y) as the response variable. A scatterplot of the data (with regression line included), and a summary of the data are provided. One of the menu it is a hamburger with 107 grams of fat and 1410 calories. Suppose we rejected the hypothesis Hoof zero slope in simple linear regression. Indicate which of the following is the only true statement.
R= correlation between x and y = 0.979
X- = mean of the values of x = 40.35 grams
? = mean of the values of y = 662.88 calories
Sx= standard deviation of the values of x = 27.99 grams
Sy= standard deviation of the values of y = 324.90 calories
o A. for every increase in one gram of fat, we expect the associated number of calories to increase by 11.36 o
oB. the rejection of I-10 suggests that there is no linear relationship between x and y
C. from the scatterplot. it seems that the estimated intercept is positive
0 D. the y variables do not need to follow a normal distribution
0 E. computer software can easily provide a confidence interval for the true slope
0 F. none of A-E are true
0 G. two of A-E are true
0 H. three of A-E are true
Question 20
Consider a multiple linear regression model where the response variable Y is the weight of a snake. and the explanatory variables are the age of the snake in months X1. a climatic variable X2 (which is coded as 0 for a cold climate and 1 for a hot climate) and the species X3 (which is coded as 0 for the common species and 1 for the exotic species). I believe that the birth weight of snakes is identical for both species in the same climatic condition. I also believe that high temperatures decrease the weight of the common species more than the weight of the exotic species. Given the physical understanding of these snakes, which model is the most appropriate?
0 Y= β0 + β1x1 + β2X3+ β3 X3 + e
0 Y= β0 + β1x1 + β2X3+ β3 X3 + β4 X2 X3+E
0 Y= β0 + β1x1 + β2X3+ β3 X3 + β4 X1 X3+E
0 Y= β0 + β1x1 + β2x1X2+ β3 X3 + β4 X1 X3+ β5 x1x2x3+E
0 Y= β0 + β1x1 + β2x1X2+ β3 X3 + β4 X1x2 X3+ e
0Y= β0 + β1x1 + β2X2+ β3 X3 + β4 X1x2 X3+ β5 x1x2+ β6x2x3+E
0 Y= β0 + β1x1 + β2x1x2+ β3 x1X3 + β4 X1x2 X3+ e
O H. none of the above
