Answer:
a) P = 0.274925
b) required confidence interval = (0.2705589, 0.2793344)
c) FALSE
d) FALSE
e) TRUE
f) There is still probability that he would win. And it would be highly unusual if he wins assuming that the true population proportion is 0.274925.
Step-by-step explanation:
a)
PROBABILITY
since total number of simulations is 40,000 and and number of times Donald Trump wins an Electoral College majority in the 2020 US Presidential Election is 10,997
so the required Probability will be 10,997 divided by 40,000
P = 10997 / 40000 = 0.274925
b)
To get 95% confidence interval for the parameter in question a
(using R)
>prop.test(10997,40000)
OUTPUT
1 - Sample proportion test with continuity correction
data: 10997 out of 40000, null probability 0.5
x-squared = 8104.5, df = 1, p-value < 2.23-16
alternative hypothesis : true p ≠ 0.5
0.2705589 0.2793344
sample estimate
p
0.274925
∴ required confidence interval = (0.2705589, 0.2793344)
c)
FALSE
This is a wrong interpretation of a confidence interval. It indicates that there is 95% chance that the confidence interval you calculated contains the true proportion. This is because when you perform several times, 95% of those intervals would contain the true proportion but as the confidence intervals will vary so you can't say that the true proportion is in any interval with 95% probability.
d)
FALSE
Once again, this is a wrong interpretation of a confidence interval. The confidence interval tells us about the population parameter and not the sample statistic.
e)
TRUE
This is a correct interpretation of a confidence interval. It indicates that if we perform sampling with same sample size (40000) several times and calculate the 95% confidence interval of population proportion for each of them, then 95% of these confidence interval should contain the population parameter.
f)
The simulation results obtained doesn't always comply with the true population. Also, result of one simulation can't be taken for granted. We need several simulations to come to a conclusion. So, we can never ever guarantee based on a simulation result to say that Donald Trump 'Won't' or 'Shouldn't' win.
There is still probability that he would win. And it would be highly unusual if he wins assuming that the true population proportion is 0.274925.
is an integer, you can use induction. First show the inequality holds for 
. You have 
, which is true.
, i.e. that 
. We want to prove the statement then must hold for 
.
for the statement to be true, so we require
. Well this is obviously true, because solving the inequality gives 
. So you're done.
increases monotonically and faster than