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ADM 2304 Lecture Notes - Lecture 5: Test Statistic, Credit Manager, Standard Deviation

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13 May 2017

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University of Ottawa

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Document Summary

The distribution of the difference between two sample proportions. > since we know the distribution of the difference of two proportions we can calculate the con dence interval in exactly the same way. > the general structure of a con dence interval is (estimate (z- or t-value) (sd or se of estimate) > thus for the difference in the proportions for two independent populations, a 100(1- )% con dence interval is given by: ( p1 p2 ) z /2. If n1p1, n2p2, n1(1 - p1) and n2(1 - p2) are all greater than 10. > note that there is no need to restrict ourselves to testing p1 - p2 = 0. We can test for any non-zero difference, p1 - p2 = p0, as well in which case the test statistic is: z. Under hypothesis h0: p1 - p2 = 0, we are assuming that the proportions are actually the same thing.

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Related Questions

Never forget that even small effects can be statistically significant if the samples are large. During a three-year period, 8 of the 57 headed by men and 6 of the 35 headed by women failed.

(a) Find the proportions of failures for businesses headed by women and businesses headed by men. These sample proportions are quite close to each other. Give the p-value for the hypothesis that the same proportion of women's and men's businesses fail. (Use two-sided alternative). What can we conclude (use alpha as 0.05)?

The p-value was ___________ so we conclude that ____________.

(b) Now suppose that the same sample proportion came from a sample 30 times as large. That is 180 out of 1050 businesses headed by women and 240 out of 1710 businesses headed by men fail. Verify that the proportions of failures are exactly the same as in part (a). Repeat the test for the new data. What can we conclude?

The p-value was ___________ so we conclude that ____________.

c) It is wise to use a confidence interval to estimate the size of an effect rather than just giving a p-value. Give 95% confidence intervals for the difference between proportions of men's and women's businesses that fail for the settings of both (a) and (b).

Interval for smaller samples: ___________ to ___________.

Interval for larger samples: ___________ to ___________.

What is the effect of larger samples on the confidence interval?

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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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