# Understanding the meaning of "expected fraction" (Statistics)

• ProbablySid
In summary: If "expected" and "fraction" both make sense, then you ought to understand what "expected fraction" means.Maybe it has to see with selecting an estimator, I guess for a proportion here.
ProbablySid
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?

ProbablySid said:
I don't know what it means by "expected fraction"? Is that just the expected value/expectation?
Can you think of anything else it could possibly mean?

PeroK said:
Can you think of anything else it could possibly mean?
I'm not really sure - it's the fraction bit that's tripping me up, and I can't find much information on what an "expected fraction" is. From a post on statistics stack exchange, someone asked a question regarding "expected fractions" and from what I could gather, it was referring to essentially how many of the observations (or in this case votes) would remain in the the top x% after a shock (in this case, the actual referendum perhaps?). I'm still quite confused and I don't know where to start.

ProbablySid said:
I'm not really sure - it's the fraction bit that's tripping me up, and I can't find much information on what an "expected fraction" is. From a post on statistics stack exchange, someone asked a question regarding "expected fractions" and from what I could gather, it was referring to essentially how many of the observations (or in this case votes) would remain in the the top x% after a shock (in this case, the actual referendum perhaps?). I'm still quite confused and I don't know where to start.
If "expected" and "fraction" both make sense, then you ought to understand what "expected fraction" means.

Maybe it has to see with selecting an estimator, I guess for a proportion here.

ProbablySid

## 1. What is the definition of "expected fraction" in statistics?

The expected fraction, also known as the expected value or mean, is a measure of central tendency in statistics. It represents the average value that would be obtained if a random experiment were repeated an infinite number of times.

## 2. How is the expected fraction calculated?

The expected fraction is calculated by multiplying each possible outcome by its probability and then summing all the products. This can be represented mathematically as E(X) = ∑xP(x), where E(X) is the expected fraction, x is the possible outcome, and P(x) is the probability of that outcome.

## 3. What is the significance of the expected fraction in statistics?

The expected fraction is a useful tool in statistics as it provides a single value that represents the most likely outcome of a random experiment. It can also be used to compare different data sets and make predictions about future outcomes.

## 4. How does the expected fraction differ from the actual fraction?

The expected fraction is a theoretical value based on probability, while the actual fraction is the observed value in a specific sample or experiment. The actual fraction may differ from the expected fraction due to chance or other factors.

## 5. Can the expected fraction be negative?

Yes, the expected fraction can be negative if the possible outcomes have negative values and their probabilities are such that they outweigh the positive outcomes. However, in most cases, the expected fraction is a positive value as it represents the average of all possible outcomes.

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