Question

A survey of 848 Americans found that 441 do not believe President Trump’s assertion that there is a crisis on the Mexican-US border.

(https://www.independent.co.uk/news/world/americas/us-politics/trump-shutdown-latestborder-wall-poll-support-tweets-republicans-democrats-graham-pelosi-a8726206.html)

(a) As a Democrat, you believe that a majority of Americans do not believe Trump’s assertion of a crisis at the US-Mexican border and want to confirm your belief. Perform a test at the 0.05 level of significance using a manual calculation of the appropriate (1-sided) confidence interval for the proportion of Americans who do not believe Trump. What is your conclusion?

(b) Suppose you wanted to take a large enough sample size to enable you to conclude with the same level of significance that more than 50% of Americans do not believe Trump’s assertion of a crisis. What sample size would be required?

(c) To test whether the proportion of college students who do not believe Trump exceeds 50%, you found 13 out of a small sample of 15 randomly selected college students who do not believe Trump. Perform the test, using the 0.05 level of significance and show how you would calculate the p-value for this test.

Answer 1

a) Sample size, n - 848 sample proportion, p =-=-~ 0.52 Level of significance, a - 0.05 At α 0.05 , a one-tailed critical value from the z-table, Z-1 .645 Compute lower bound for the population proportion (p) ,x 441 n 848 20.52-(1.645)V (0.52)(1-0.52) 848 2 0.52 -0.028222 0.491778 Thus, the lower confidence bound is 0.49 (approximately) As, lower bound is less than 0.5, there is no sufficient evidence to conclude that the more than 50% of Americans do not believe Trump's assertion of a crisis b) At α-0.05 , a one-tailed critical value from the z-table, Ζα-1 .645 From part (a), sample proportion and lower bound are Sample proportion, p=0.52 Lower confidence bound 0.49 Margin of error, E- p Lowerbound 0.52 -0.49 0.03

Compute sample size (0.52) (1-0.52) 1.645 0.03 750 Thus, the required sample size should be, 750 c) Null hypothesis, Ho p-0.5 Alternative hypothesis, H p>0.5 Level of significance, a - 0.05 Decision rule: Reject H,if p-value 3 0.05 Sample size, n=15 As, sample size is very small, you will use binomial distribution for calculating p-value Let Xdenotes number of students who do not believe Trump Here, X, ~ Binom(n=15,p=0.5) The probability function of Xcan be written as, 15 15 0.5) x= 0.1.2 15 From the rejection region can be defined as. Reject Hoif X>13, otherwise fail to reject Ho

Compute the p-value as follows: p-value = P(X > 13 | p = 0.5) -P(X=14)-P(X=15) 15-14 15 15(o.5)"(1-0.5) 5 5 (s)s) 0.000488281 0.0005 Since p-value is less than 0.05, reject the null hypothesis, Ho at a 5% significance level. There is sufficient evidence to conclude that the proportion of college students who do not believe Trump exceeds 50%.