- #1

ProbablySid

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- Homework Statement
- I am trying to do a statistics question from a synoptic practise exam paper. The gist of the question is that a poll of sample size = 2091 people was taken a few days prior to the 2016 referendum in the UK on whether to remain or leave the EU. 1062 people said they intended to vote remain, and 1029 said they intended to vote leave (undecided voters were ignored). In a simple model, this can be treated as a binomial process with independent voters who have a 50% chance of voting remain, and 50% chance of voting leave. I was able to do some parts of the question, which I will explain in my attempts, but I am now stuck on a question asking about an "expected fraction", which again I will explain more about in my attempt.

- Relevant Equations
- Expectation value of a binomial distribution: ##E(X) = \mu = Np##

Standard deviation of a binomial distribution: ##\sigma = (Np(1-p))^{\frac{1}{2}}##

z-score of a Gaussian distribution: ##z = \frac{X-\mu}{\sigma}##

The first part of the question asked me to calculate the mean and standard deviation for the number of remain votes in the simple binomial model consisting of total sample size of 2091 people. I believe this is fairly straightforward, it was simply ##E(X) = \mu = 2091(0.5) = 1045.5## votes and for the standard deviation, similar application of the relevant formula which gave ##\sigma \approx 22.86## votes.

The next part of the question asked me to calculate for this binomial model that the pre-referendum poll would produce a number of remain votes equal or greater than 1062. For this question, I used a Gaussian model to approximate the binomial distribution, as the value of N is very large. The Gaussian was modeled as having ##\mu = 1045.5## and ##\sigma = 22.86##. Then, using the z-score formula:

$$z = \frac{1061 - 1045.5}{22.86} \approx 0.68 \approx 0.7$$

And then the exam paper provided a z score table and ##z = 0.7## corresponded to ## \int_{0.7}^\infty P(X)dX = 0.24##, thus

##P(X \geq 1062) \approx 1-0.24 = 0.76##

Now the part I am stuck on. The next part of the question asks to consider a binomial model of independent voters based on probabilities suggested by the outcome of the pre-referendum poll (assuming that means using the previous parts I've calculated). It now asks me to "calculate the expected fraction of remain votes and the standard deviation of this fraction from this model."

I don't know what it means by "expected fraction"? Is that just the expected value/expectation? I've tried to look through my statistics lecture notes from back in first year, but there is no mention of an "expected fraction". Is it simply referring to the fact that the expectation value cannot be known since we don't know the sample size for this new model (?) and thus we should just express it as a fraction?

The next part of the question asked me to calculate for this binomial model that the pre-referendum poll would produce a number of remain votes equal or greater than 1062. For this question, I used a Gaussian model to approximate the binomial distribution, as the value of N is very large. The Gaussian was modeled as having ##\mu = 1045.5## and ##\sigma = 22.86##. Then, using the z-score formula:

$$z = \frac{1061 - 1045.5}{22.86} \approx 0.68 \approx 0.7$$

And then the exam paper provided a z score table and ##z = 0.7## corresponded to ## \int_{0.7}^\infty P(X)dX = 0.24##, thus

##P(X \geq 1062) \approx 1-0.24 = 0.76##

Now the part I am stuck on. The next part of the question asks to consider a binomial model of independent voters based on probabilities suggested by the outcome of the pre-referendum poll (assuming that means using the previous parts I've calculated). It now asks me to "calculate the expected fraction of remain votes and the standard deviation of this fraction from this model."

I don't know what it means by "expected fraction"? Is that just the expected value/expectation? I've tried to look through my statistics lecture notes from back in first year, but there is no mention of an "expected fraction". Is it simply referring to the fact that the expectation value cannot be known since we don't know the sample size for this new model (?) and thus we should just express it as a fraction?