In a CNN poll conducted in July 2016, a random sample of 882 registered voters i

Question

In a CNN poll conducted in July 2016, a random sample of 882 registered voters in the IIS were asked whether they thought Hillary Clinton is "honest and trustworthy" as well as whether they thought Donald Trump is "honest and trustworthy". The following data were recorded. (a) Construct a 95% confidence interval for the proportion, p_1, of US voters who would have responded that Hillary Clinton was honest and trustworthy if they had been surveyed in July 2016. (b) State the name of the theorem that allows us to assume the sampling distribution of p_1 is approximately normal (and therefore allows us to construct the confidence interval in part (1)a)? (c) What range of values could the population proportion p_1 take so that the assumption of an approximately normal sampling distribution for p_1 is valid? (d) Test the hypothesis that the proportion of voters who considered Hillary Clinton to be honest and trustworthy is the same as the proportion of voters who considered Donald Trump to be honest and trustworthy in July 2016, at the 5% level of significance. Your answer should include: (i) a statement of the null and alternative hypotheses (including an explanation of any parameters used), (ii) the formula for the test statistic, (iii) the reference distribution of the test statistic (if H_0 is true), (iv) the value of the observed test statistic, (v) the associated P-value, and (vi) whether the null hypothesis should be retained or rejected. (e) State the conclusion of the hypothesis test in context and without reference to the technical language of hypothesis testing. (f) Find the margin of error for the 95% confidence interval for p_1, (see (1)a) in the (worst) case where p_1 = 0.5. (g) What is the minimum sample size n*_1 required to halve the margin of error in (1)f, still assuming p_1 = 0.5? Comment on n*_1 compared to n_1.

Explanation / Answer

Solution:

a. p1 = 268/882 = 0.304

a = 1 – 0.95 = 0.05

Using Z-tables, the critical value is

Z (a/2) = Z (0.05/2) = Z (0.025) = ±1.96

95% confidence interval is given by:-

p ± Z*p (1 – p)/n

0.304 ± 1.96*0.304*(1 – 0.304)/882

0.304 ± 1.96*0.015

0.304 ± 0.030

0.304 – 0.030, 0.304 + 0.030

0.273, 0.334

b. Central limit theorem.

c. The range of values is 27.3% to 33.4%

d.

i. Null Hypothesis (Ho): p1 = p2

Alternative Hypothesis (Ha): p1 p2

Where p1 is the population proportion of US voters, who would have responded that Hillary Clinton was honest and trustworthy,

p2 is the population proportion of US voters, who would have responded that Donald Trump was honest and trustworthy

Pooled proportion, p = (x1 + x2)/(n1 + n2)

Pooled proportion, p = (268 + 380)/(882 + 882)

Pooled proportion, p = 0.367

p1’ = 268/882 = 0.304, p2’ = 380/882 = 0.431

ii. Test Statistics

Z = (p1’ – p2’)/p (1 – p) (1/n1 + 1/n2)

iii. Z-distribution is used because sample size is large and hence it approximates to normal distribution.

iv. Test Statistics

Z = (p1’ – p2’)/p (1 – p) (1/n1 + 1/n2)

Z = (0.304 – 0.431)/0.367*(1-0.367) (1/882 + 1/882)

Z = -5.53

v. Using Z-tables, the p-value is

P [Z > 5.33] + P [Z < -5.33] = 0.000

vi. Since p-value is less than 5% level of significance, we reject Ho.

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