Question

According to data released by FiveThirty Eight (data drawn on Monday, August 17th, 2020), Donald Trump wins an Electoral College majority in the 2020 US Presidential Election in 10,997 simulations out of a total of 40,000 simulations.

(a) I am interested in the probability that Donald Trump wins any one simulation from the Five Thirty Eight model. Write the parameter of interest. Write and calculate the estimate of your parameter using mathematical notation.
(b) Construct a 95% confidence interval for the parameter from (a), show at least 4 significant digits. I highly recommend using R, and only rounding at the end.
(c) TRUE or FALSE:
According to the FiveThirty Eight model, there is a 95% probability that the true parameter of interest from
(a) lies within your interval from (b).
(d) TRUE or FALSE:
If I were to repeat samples of 40,000 from the Five Thirty Eight model, I expect 95% of my sample statistics from (a) to lie within your interval from (b).
(e) TRUE or FALSE:
If I were to repeat samples of 40,000 from the FiveThirty Eight model, I expect 95% of samples to lead to 95% confidence intervals that cover the true parameter of interest from (a).
(f) People often misinterpret the FiveThirty Eight model to say that our previous results suggest that Donald Trump "shouldn't" or "won't" win the election. Explain why this is not a correct interpretation. Suggest, calculate, and explain an alternative conclusion from the simulation results that is more intuitive and interpretable.

Answer 1

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.

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.