1000 STAT Assignment 2 DUE: February 8th (Wed. Eve. Section), February 9th (T/Th. Sections), February 10th (MWF. Sections) SHOW ALL YOUR WORK
[5] 1. Consider the following 10 data points: x y 3 4 5 3 6 2 4 1 3 2 7 3 6 3 5 5 4 4 7 2
(a) Plot the above observations on a scatterplot.
untitled: Fit Y by X of y by x
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Solution:
Bivariate Fit of y By x
5 4 3 2 1 3 4 5 x 6 7 y
Linear Fit
Linear Fit
(b) Calculate the correlation coefficient, r (by hand ). What does this value tell us?
Summary of Fit
y = 3.4 - 0.1*x
Solution: Preliminary x = 5 and y sx = 1.49 and sy = 1.20 Analysis of Variance r= 1 n-1
n
RSquare RSquare Adj Root Calculations: Mean Square Error Mean of Response = 2.9 Observations (or Sum Wgts)
0.015504 -0.10756 1.25996 2.9 10
i=1
xi - x sx
Source yi Model - Error s y C. Total
y
Sum of DF Squares n Mean Square F Ratio 1 1 0.200000 0.20000 0.1260 =8 12.700000 (xi - x) (yi - y ) (n - 1)sx sy i=1 1.58750 Prob > F 9 12.900000 0.7318
Parameter Estimates
xi - x (3 - 5) = -2 (5 - 5) = 0 (6 - 5) = 1 (4 - 5) = -1 (3 - 5) = -2 (7 - 5) = 2 (6 - 5) = 1 (5 - 5) = 0 (4 - 5) = -1 (7 - 5) = 2
yi -Term y (4 - 2.9)Intercept = 1.1 x (3 - 2.9) = 0.1 (2 - 2.9) = -0.9 (1 - 2.9) = -1.9 (2 - 2.9) = -0.9 (3- 2.9) = 0.1 (3 - 2.9) = 0.1 (5 - 2.9) = 2.1 (4 - 2.9) = 1.1 (2 - 2.9) = -0.9
(xi - Error Estimate x)(yi - y ) t Ratio Std 3.4 -2.2 1.463942 2.32 -0.1 0.281736 -0.35 0 -0.9 1.9 1.8 0.2 0.1 0 -1.1 -1.8 n i=1 (xi - x) (yi - y ) = -2
Prob>|t| 0.0487* 0.7318
Therefore
-2 = -0.124 (9)(1.49)(1.20) Interpretation: There is weak negative association between x and y.
(c) Calculate the least squares regression line. Solution: Preliminary Calculations: x = 5 and y = 2.9 sx = 1.49 and sy = 1.20 and r = -0.124 (from the previous question), if it is used correctly in this part and was wrong in the previous part, no additional marks are deducted. b1 = r sy sx = -0.124 1.20 1.49 = -0.10
b0 = y - b1 x = 2.9 - (-0.10)(5) = 3.4 Therefore, y = 3.4 - 0.10x ^
[11] 2. A foods-and-nutrition investigator would like to know if the amount of sodium (g) can be predicted by the amount of sugar (g) in the following sample of 12 popular breakfast cereals. Cereal Sugar (g) Mini Wheats 7 All-Bran 5 Apple Jacks 14 Captain Crunch 12 Cheerios 1 Cinnamon Toast Crunch 13 Corn Flakes 2 Raisin Bran 12 Oat Bran 10 Crispix 3 Frosted Flakes 11 Fruit Loops 13 (a) Identify the response and explanatory variables. Sodium (g) 0 0.26 0.13 0.22 0.29 0.21 0.29 0.21 0.14 0.22 0.20 0.13
Solution: The response variable is the amount of sodium. The explanatory variable is the amount of sugar. (b) Use JMP to produce a scatterplot and highlight the least squares regression line on the output.
cereals: Fit Y by X of SODIUM(g) by SUGAR(g)
Solution:
0.3 0.25 0.2 SODIUM(g) 0.15 0.1 0.05 0 -0.05 0
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Bivariate Fit of SODIUM(g) By SUGAR(g)
2.5
5
7.5 10 SUGAR(g)
12.5
15
Linear Fit
Linear Fit
SODIUM(g) = 0.2581201 - 0.0077422*SUGAR(g)
Summary of Fit
RSquare RSquare Adj Root Mean Square Error Mean of Response Observations (or Sum Wgts) 0.201732 0.121905 0.076529 0.191667 12
Analysis of Variance
Sum of Source DF Squares Mean Square F Ratio (c) Interpret the slope of the least squares regression line in the context of this question. Model 1 0.01480042 0.014800 2.5271
Solution: Estimates Parameter Term Estimate Std ForIntercept one gram Error t Ratio in0.0003* we PREDICT sodium to decrease by every 0.2581201 0.047281 5.46 Prob>|t| increase sugar SUGAR(g) -0.007742 0.00487 -1.59 0.1430 0.008 grams. (d) What is the percentage of variation in sodium that is explained by its regression on sugar? You may use your JMP output. Solution: This is r2 = 0.201, therefore, 20.1% of the variation in sodium is explained by its regression on sugar. (e) What is the residual for Captain Crunch? Solution: Residual = Observed - Predicted Residual = y - y ^
Error C. Total
10 11
0.05856625 0.07336667
0.005857
Prob > F 0.1430
The observed value, y for Captain Crunch is 0.22 grams of Sodium The predicted value, y for Captain Crunch is y = 0.258 - 0.008(12) = 0.162 ^ ^ The residual is: 0.22 - 0.162 = 0.058 (f) Using your JMP output, what is the value of the correlation coefficient? Solution: The output gives us R2 = 0.201, r = - 0.201 = -0.448 Make sure there is a negative because there is negative association at X increases, Y decreases. (g) Suppose we calculated sodium and sugar in milligrams (mg) instead of grams (g). What effect would that have on the correlation coefficient? Solution: There would be no effect because r is unit-less. (h) Is Mini Wheats an outlier or an influential observation? Explain. (no calculations required ). Solution: Mini Wheats is an outlier as it falls far beyond the rest of the observations in the y-direction. Influential observations are "outliers" in the x-direction such that removal of those observations will cause a dramatic shift in the least squares regression line. (i) What is the predicted sodium content of a cereal with 23 grams of sugar? Is this prediction reliable? Explain. Solution: y = 0.258 - 0.008(23) = 0.074. This prediction is not reliable because it is ^ extrapolation. The least squares regression line is only reliable for the range of values that was used to construct it.
[16] 3. You have been asked to develop an experiment to meet objectives the for each of the following scenarios. For each of the studies, answer the following questions:
i. What is the appropriate type of experimental design that should be used? ii. What are the experimental units? iii. What is/are the factor(s) under investigation? iv. What are the factor levels?
v. What are the treatments to be applied? vi. Is there a blocking variable? If so, what is it? vii. Draw a chart to outline the proposed design of the experiment.
(a) A tire manufacturer would like to determine how weather conditions and speed affect the stopping distance of a vehicle using a certain type of tire. An experiment will be conducted on a test track under simulated weather conditions (either dry, rain or snow) and at different speeds (40 km/h, 60 km/h, 80 km/h or 100 km/h). Each combination of factor levels will be tested in three trial runs. Solution: i. The most appropriate type of design in this situation is a completely randomized design. ii. The experimental units are the tires. iii. The factors are weather conditions and speeds. iv. There are three levels to factor A (dry, rain or snow) and four levels of factor B (40 km/h, 60 km/h, 80 km/h, 100 km/h). v. The treatments are all combinations of factors and levels: ) dry and 40km/hr. ) dry and 60km/hr. ) dry and 80km/hr. ) dry and 100km/hr. ) rain and 40km/hr. ) rain and 60km/hr. ) rain and 80km/hr. ) rain and 100km/hr. ) snow and 40km/hr. ) snow and 60km/hr. ) snow and 80km/hr. ) snow and 100km/hr. vi. There is no blocking variable.
vii. (b) An experiment is to be conducted to determine the effect of temperature on the adhesive strength of C-glue. The glue will be applied to a material at either 10 C, 20 C or 30 C. The experiment will be conducted separately for two different types of material (metal and plastic), as the effect of temperature is expected to differ for the two materials. Ten pieces of metal and ten pieces of plastic will be tested for each treatment. Solution: i. The most appropriate type of design in this situation is a randomized block design. ii. The experimental units are types of material. iii. The factor is temperature. there is only one factor. iv. There are 3 levels: 10 C, 20 C or 30 C. v. Since there is only one factor, treatments are just 10 C, 20 C or 30 C. vi. There is one blocking variable, and the blocking variable is type of material metal or plastic.
vii.
[4] 4. A running-shoe manufacturer wanted to test the effect of its new sprinting shoe on the 100-meter dash times. The company sponsored five athletes who were running the 100meter dash in the Summer Olympics. To test the shoe, it had all five runners run the race with its competitor's shoe and then again with its new shoe. The company used the difference in times as its response variable. (a) Suggest some improvements to this design.
Solution: First, they are using athletes that they have sponsored so that might result in bias. They should have randomly selected athletes to test the shoe. They should have also randomized which race will be run with which shoe. They had all the runners first run with the competitor's shoe and then the new shoe. They could have not told the athlete which shoe they were wearing to run the race (blinding). They should also replicate the experiment several times since running times will vary under different shoe conditions. (b) Why might the shoe manufacturer not be able to generalize the results they found to all runners? Solution: Because of the issues in (a), they results may systematically favour their shoe. In addition this was done on athletes, these results may not be able to be generalized for the general runner.
[2] 5. In September 1998, USA Weekend magazine asked, "Should humans be cloned?" Readers were invited to register a "Yes" or "No" answer by calling one of two different 800 numbers. Based on 38 023 responses, the magazine reported that "9 out of 10 readers oppose cloning." Is this conclusion justified? Why or why not? If not, describe the types of bias that may be present.
Solution: The conclusion is not justified. This is a voluntary response sample, only people who feel strongly about the topic will phone the 800 number and voice their opinion. Undercoverage is also prevalent here because no everyone in the population will read this magazine so this isn't a fair representation of the entire population.
[2] 6. There are 20 first-class passengers and 120 coach passengers scheduled on a flight. In addition to the usual security check, 12 passengers will be subjected to a more complete search. (a) Describe a sampling strategy to randomly select those to be searched. Solution:
We would like to have a stratified random sample, so we can choose passengers from both coach and first-class. (b) Here is the first-class passenger list and a set of random digits. Select two passengers to be searched, carefully demonstrating your process. 65436 71127 04879 41516 20451 02227 917469 23593
Bergman Cox Bowman DeLara Burkin Dell Castillo Dugan Clancy Febo
Fontana Forester Frongillo Furnas LePage
Perl Rabkin Rofli Swafford Testut
Solution: The passengers will be labeled 01 to 20, The passengers selected will be: Fontana and Castillo. If you labelled the passengers from 00 to 19, The passengers selected will be: Forester and Clancy.