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STAT 301 Lecture Notes - Lecture 20: Scatter Plot, Regression Analysis, Simple Linear Regression

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7 Nov 2017

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Benjamin Prytherch

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Document Summary

Often it is of interest to see not just how strong x and y are correlated, but also how they. To answer this, we use a technique called linear regression, which will be introduced in the next set of notes. Note that when r is fairly near zero, it is very difficult to tell that the data are correlated just by looking. We can answer this using a hypothesis test. If we reject h0: p = 0, we call our correlation significant Remember that statistically significant results are those that are large enough to be unlikely to occur by chance, assuming h0 is true. So, a significant sample correlation (r ) is one that is larger than we would expect, if the population correlation (p) was zero. Also, remember that larger sample sizes (n) make it easier to reject h0, because the standard error of a statistic goes down as n goes up.

Related textbook solutions

Introductory Statistics

OER Edition, 2013

ISBN: 9781938168208

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Related Questions

A random sample of soil specimens was taken from a large geographic area. The specimens can be assumed to be independent. The amount of organic matter, as a percent, was determined for each specimen. The data are below:
0.14, 0.32, 1.17, 1.45, 3.5, 5.02, 5.09, 5.22
A soil scientist wants to know whether the population mean percent organic matter is different than 4%. A
significance level of α = 0.05 is chosen.

1. (a) State hypotheses appropriate to the research question.
2. (b) Graph the data as you see fit. Why did you choose the graph(s) that you did and what does it (do they) tell you?
3. (c) Regardless of your conclusion from (b), use the bootstrap to perform a test of the hypotheses you stated in (a). Use B = 8000 resamplings. Compute the p-value, and make a reject or not reject conclusion. Then state the conclusion in the context of the problem. In other words, does it seem the mean organic matter level is different than 4%?
4. (d) Regardless of your conclusion from part (b), use a T -test to perform a test of the hypotheses you stated in (a). Compute the p-value, and make a reject or not reject conclusion. Then state the conclusion in the context of the problem. In other words, does it seem the mean organic matter level is different than 4%?. (I recommend doing this part with a hand calculator and statistical tables as practice for exam conditions, but you may check your answers using R if you wish.)
5. (e) Compare your answers from parts (c) and (d). Which method do you think is better? Are you surprised at the similarity or dissimilarity? What do you think explains this?

(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,

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

include columns for Source, df, SS, MS and F, and should include rows for the model

(full model), residual and total.

(c) (2 marks) In addition to the full model, the reduced model that includes only an

intercept and age is ﬁt. What is the null hypothesis being considered by ﬁtting these

two models? (Refer to quantities deﬁned in your model.)

(d) (3 marks) The SSM from the reduced model is 59.7. Calculate an F-statistic to test the hypothesis of part (c). State the degrees of freedom for the F-distribution

you would compare this statistic to. (You do not need to calculate the p-value or actually test the hypothesis.)

(e) (1 mark) Using terminology from class, name the kind of Ftest that the statistic in part (d) corresponds to.

(f) (1 mark) If the null model for these data is compared to the model that includes an

intercept and age, what are the results?

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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

(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.

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