STAT3012 Lecture Notes - Lecture 5: Standard Deviation, Forb, Sampling Distribution

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23 Jun 2018

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Lecture 5 - Inference and prediction in simple linear regres-

sion

New concepts

✷Hypothesis testing for β0and β1

✷Conﬁdence intervals for β0and β1

✷Conﬁdence interval for σ2

✷confint() command

Applied Linear Models: Lecture 5 1

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Theory – Sampling distribution of b

β1

✷Standardizing the normal sampling distribution of b

β1gives

β1−β1

σ/√Sxx ∼N(0,1).(1)

✷Replacing σ2with the estimate s2=RSS/(n−2) changes the sampling distri-

bution of b

β1(for a proof see Kutner et al; 2005, p45):

β1−β1

SE(b

β1)∼tn−2,(2)

where

◦the notation tn−2signiﬁes a t-distribution with (n−2) degrees of freedom

(df);

◦the term SE(b

β1) = s

√Sxx is the standard error of b

β1; i.e. the (estimated)

standard deviation of the sampling distribution of b

β1.

Applied Linear Models: Lecture 5 2

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Theory – Sampling distribution of b

β1

✷Equation (2) forms the basis for testing hypotheses and constructing conﬁdence

intervals for β1.

✷For testing H0:β1=β0

1, against H1:β16=β0

1, use the t-test statistic

t=b

β1−β0

SE(b

β1).(3)

✷Find the p-value corresponding to t. As usual, smaller p’s indicate more/stronger

evidence against H0.

✷A(1 −α)conﬁdence interval for β1is

β1±tn−2(1 −α/2) ×SE(b

β1).(4)

Applied Linear Models: Lecture 5 3

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Document Summary

Lecture 5 - inference and prediction in simple linear regres- sion. Standardizing the normal sampling distribution of b 1 gives b 1 1. Replacing 2 with the estimate s2 = rss/(n 2) changes the sampling distri- bution of b 1 (for a proof see kutner et al; 2005, p45): b 1 1. Theory sampling distribution of b 1 intervals for 1. Equation (2) forms the basis for testing hypotheses and constructing con dence. For testing h0 : 1 = 0. 1, against h1 : 1 6= 0. Find the p-value corresponding to t. as usual, smaller p"s indicate more/stronger evidence against h0. A (1 ) con dence interval for 1 is b 1 tn 2(1 /2) se(b 1) . With similar steps we get that a 100(1 )% con dence interval for 0 is. Theory con dence interval for b 0 where the standard error of b 0 is given by b 0 tn 2(1 /2)se(b 0),

Related Questions

2. (12 marks) In a study of 85 individuals, food questionnaires were used to measure the change in polyunsaturated fatty acids (PUFA) intake from the beginning to the end of the study. In addition researchers measured a genetic marker in the TNF-αgene; for this analysis, we will take the two values of this TNF-αvariable to be A- or A+. The researchers also collected information on gender, age and BMI. The primary question of interest is whether TNF-αmodiﬁes the eﬀect of PUFA on HDL.

(a) (1 mark) Let TNFa be a variable with value 1 for A+ and 0 for A- TNF-αstatus and

let gender be 1 for males and 0 for females. State a linear model for mean change in HDL that includes the predictors PUFA, TNFa, PUFA×TNFa interaction, and eﬀects for gender, age and BMI.

(b) (1 mark) A summary of the ﬁtted model is:

Estimate Std. Error t value Pr(>|t|)

(Intercept) 0.037531 0.196980 0.191 0.8494

pufa 0.019143 0.009245 2.071 0.0417

TNFa 0.074433 0.034128 2.181 0.0322

gender -0.008988 0.031793 -0.283 0.7781

age -0.003895 0.002156 -1.806 0.0747

bmi 0.006777 0.004332 1.565 0.1217

pufa:TNFa -0.024575 0.017844 -1.377 0.1724

Write a short sentence to describe why we would even consider removing gender from the model (i.e., why we would not just be happy with the above model from

the outset).

(c) (2 marks) Below is a summary of the model ﬁt without gender. Using our 10% rule-of-thumb, is there any evidence that gender confounds the PUFA×TNFa eﬀect?

If so why, if not why not?

Estimate Std. Error t value Pr(>|t|)

(Intercept) 0.024467 0.190364 0.129 0.8981

pufa 0.018841 0.009130 2.064 0.0423

TNFa 0.074850 0.033897 2.208 0.0301

age -0.003783 0.002107 -1.795 0.0765

bmi 0.006835 0.004301 1.589 0.1161

pufa:TNFa -0.024496 0.017737 -1.381 0.1712

(d) (2 marks) The researchers decided to use the model with PUFA, TNFa, PUFA×TNFa, age and BMI for inference of PUFA×TNFa interaction. What are the degrees of

freedom for the t-distribution whose critical value we need to construct conﬁdence intervals for the interaction? What upper tail probability should you use for a 95%

conﬁdence interval?

(e) (1 mark) The appropriate critical value is 1.96. Construct a 95% conﬁdence interval for the interaction.

(f) (2 marks) State the null hypothesis and a two-sided alternative hypothesis for testing whether TNFa modiﬁes the eﬀect of PUFA on HDL.

(g) (1 mark) From the computer output, what is the p-value for this test?

(h) (2 marks) For ﬁxed age and BMI, what is the estimated eﬀect of a one-unit change in PUFA for a subject who is A- at TNF-α? For ﬁxed age and BMI, what is the estimated eﬀect of a one-unit change in PUFA for a subject who is A+ at TNF-α.

crimsonyellowjacket200

Problem 1 (2 points). Every month a clothing store conducts an inventory and calculates the losses due to theft. The store would like to reduce these losses and is considering two methods to do so: the first is to hire a security guard and the second is to install cameras. The manager of the store has conducted the experiment: During the first 6 months he hired a security guard (method 1) and during the next 6 months he installed cameras (method 2). The monthly losses were reported as follows:

Security guard 350 289 407 398 470 259 Cameras 488 301 276 380 421 425

a/ Test with a=0.05 to determine whether there is enough evidence to conclude that expected monthly loss when installing cameras differs from 382. b/Since the cameras are cheaper than the guard, the manager would prefer to install the cameras unless there was enough evidence to infer that the guard was better. Conduct a hypothesis testing to advise what the manager should do? (Assuming equal variances and using a= 0.05)

Problem 2 (2 points). A supermarket chain performed a survey to help determine desirable locations for its new stores. The management of the chain wanted to know whether a linear relationship exists between weekly take home pay (s) and weekly food expenditures (y). A random sample of eight households produced the data shown in the following table Household 1 3 7 8 2 4 5 6 550 450 560 340 510 710 470 660

170 100 160 80 120 190 140 140 Y

a/ Find the least square regression line.

b/Perform the test Ho: Bi=0, H4: B10 (using a = 0.05) to determine whether there is enough evidence to infer that a linear relationship exists between weekly take home pay (s) and weekly food expenditures (y).

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(14 marks) Fortune magazine publishes a list of the world’s billionaires each year. We will examine the top 97 of the 1992 list. Their wealth, age, and geographic location (Asia, Europe, Middle East, United States, and Other) are reported. Wealth is a right-skewed variable, so we will consider the natural logarithm log wealth to be the response and will see if age and geographic location are predictors.

(a) (2 marks) Let

Ebe a variable that is 1 if the person is from Europe and 0 otherwise,

Mbe a variable that is 1 if the person is from the Middle East and 0 otherwise,

Obe a variable that is 1 if the person is from the “Other” region and 0 otherwise,

and

Ube a variable that is 1 if the person is from the United States and 0 otherwise.

State a linear model for the mean of log wealth that includes age and the above 4

geographic variables. Call this the full model.

(b) (5 marks) The SSY for these data is 28.3351 and the SSE from ﬁtting the full model

is 27.0461. Use this information to construct an ANOVA table. Your table should