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STAT 2050 Study Guide - Midterm Guide: Independent And Identically Distributed Random Variables, Null Hypothesis

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15 Apr 2018

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Unequal variances, how different are the samples: sew( (cid:2869) (cid:2870)(cid:4667)= (cid:3117)(cid:3118)(cid:3041)(cid:3117)+(cid:3118)(cid:3118)(cid:3041)(cid:3118) (cid:4666)(cid:3117)(cid:3118)(cid:3117)+(cid:3118)(cid:3118)(cid:3118)(cid:4667)(cid:3118) (cid:3117)(cid:3117) (cid:3117)(cid:4678)(cid:3117)(cid:3118)(cid:3117)(cid:3118)(cid:4679)+ (cid:3117)(cid:3118) (cid:3117)((cid:3118)(cid:3118)(cid:3118)(cid:3118), tobs= (cid:3117) (cid:3118) (cid:3116) Type 1: reject the h0 when it is true. P-value is very small (<0. 05 type 2: fail to reject the null when null is false p-value is large (>0. 05) 95% ci interpretation: we are 95% confidence that the true difference of the 2 means is between ( , ). One mean, mu and variance are unknown one sample t-test assumptions: observations are independent and identically distributed (iid) and randomly sampled. Threshold approach: is there actually a difference: state the hypothesis: (cid:4667) (cid:4666)(cid:3117) (cid:3118) (cid:3043) (cid:3117)(cid:3117)+(cid:3117)(cid:3118) ~(cid:1872)(cid:3041)(cid:3117)+(cid:3041)(cid:3118) (cid:2870) (cid:1873)(cid:1866)(cid:1856)(cid:1857)(cid:1870) (cid:1872) (cid:1857) (cid:1866)(cid:1873) (cid:1868)(cid:1867)(cid:1872) (cid:1857)(cid:1871)(cid:1871) (cid:3117) (cid:3118: (pooled) (cid:1871)(cid:1868)= (cid:4666)(cid:3041)(cid:3117) (cid:2869)(cid:4667)(cid:3117)(cid:3118)+ (cid:4666)(cid:3041)(cid:3118) (cid:2869)(cid:4667)(cid:3118)(cid:3118) (cid:3041)(cid:3117)+(cid:3041)(cid:3118) (cid:2869, 95% ci for 2 sided: (cid:2869) (cid:2870) + (cid:1872)(cid:2868). 975, (cid:1871)(cid:1868)(cid:4666) (cid:2869)(cid:3041)(cid:3117)+ (cid:2869)(cid:3041)(cid:3118), h0: =0 vs ha: 0.

Related textbook solutions

Introductory Statistics

OER Edition, 2013

ISBN: 9781938168208

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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

(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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