# When is a sample proportion p hat instead of x bar

I just started my first statistics class and am not majoring in statistics so sorry if this sounds like a beginner question and also sorry if my language is incorrect. (feel free to correct me.) I have been learning about creating sample distributions of phat and also sample distributions of xbar. I was wondering if you can tell the difference between when one is needed and when the other is needed by looking at a mean, standard deviation and sample size.

I have two examples from my class one requires a sample distribution of phat and the other a sample distribution of xbar

## First example using the sample distribution of xbar

Aamco Heating and Cooling, Inc., advertises that any customer buying an air conditioner during the first 16 days of July will receive a 25 percent discount if the average high temperature for this 16 day period is more than 5 degrees above normal. Daily high temperatures in July are normally distributed with a mean of 84 degrees and a standard deviation of 8 degrees.

If we consider the first 16 days of July to be a random sample, what are the expected value, standard deviation, and shape of the sampling distribution of the sample mean? (don't answer this question it's just here to show the question in context.)

## And now the second using the sample distribution of phat

Assume that 30% of all business students at a university invest in the stock market. We randomly pick 500 students

Show the sampling distribution of phat, the sample proportion of business students at this university who invest in the stock market. (Yet again no need to do this just giving context.)

So yet again I'm just asking if there is a way to tell if I need to use the equations for xbar or for phat when given a mean, standard deviation, and sample size and asked to give a sampling distribution. (And yes I know the second example says give the sampling distribution of p-hat, but I want to know if there is a way to tell if it didn't say that.) Thanks and sorry again if this is a bad question.

Here are the meanings of x bar and p hat that were used to solved the first and last question respectively:

• You need to be a little cautious about assuming that particular symbols like xbar and phat will always have the same meaning, as they are just symbols. However, those two are quite common and consistent. The first is a mean which is the sum of the observations divided by the number of observations. The second is a proportion, the number of 'successes' divided by the number of 'attempts'. You should be able to determine which type of summary is relevant by considering the type of problem you are dealing with. – Michael Lew Feb 22 '14 at 20:20
• The point in using two different notations is that, when you sample a Bernoulli variable with mean $p$, its variance is $p(1-p)$. So the variance of your estimate $\hat p$ (which is indeed the sample mean) is $\simeq {1\over n} \hat p(1-\hat p)$ ; you will use this for CI or test procedures. When you sample temperatures, you will have to use both the sample mean and the sample variance for CI and tests. – Elvis Feb 22 '14 at 22:03
• @MichaelLew so apparently this question now has over ten thousand views (which is apparently enough to make it a famous question). I felt like your comment was good and was curious if maybe you could create an answer and give helpful examples for newbies. For example you could answer the how of choosing the relevant summary based on the problem. Yes my question doesn't have many upvotes which may indicate that not many people have the same problem as I had, but as an exception there may be lots of newbies with no ability to vote on the site asking the same question. – John Dec 6 '16 at 21:18
• When you are given data and wish to draw inference about the population distribution there are more things you can estimate from the data then just the mean. That could be the reason for different notation. I don't think this us a famous question. – Michael R. Chernick Dec 6 '16 at 21:59
• @MichaelChernick Sorry, I only called it a famous question since that is the badge I received for it receiving over 10,000 views. – John Dec 7 '16 at 0:23

Both questions are essentially applications of the Central Limit Theorem, which says (informally) that "the value of a sum over many samples from a common population will tend to a normal distribution as the number of samples becomes large".

The two questions differ in the type of data that they treat. The "xbar" question concerns temperature, which is a continuous measurement (e.g. a decimal number). The "phat" question implicitly concerns a binary measurement (true/false, e.g. each student either invests or does not).

Commonly a measurement of a random variable will be denoted by $x$. For a random sample $x_1,\ldots,x_N$ the sample mean will then be denoted by $\bar{x}=\frac{1}{N}\sum_ix_i$. This applies directly to the "xbar" question. Here each $x_i$ is a temperature measurement, and the question asks about the sampling distribution of $\bar{x}$. (This arises when $\bar{x}$ is computed many times over different samples, each of size $N$).

For the "phat" question, the notation and logic is consistent with this, but the connection is a little more involved. In this case each $x_i$ will correspond to an individual student, who either invests ($x=1$) or does not ($x=0$). The probability that a student will invest would commonly be denoted by $p$ ($=30\%$ in this case). These conventions of $\Pr[\text{true}]=p$ and $\{\text{true,false}\}=\{1,0\}$ are standard for the case of a binary random variable.

Now imagine we do not know the value of $p$, but wish to estimate it from a random sample of students $x_1,\ldots,x_N$. For a single student the expected value of $x_i$ is $p$, denoted $\mathbb{E}[x]=p$ (see also here). Similarly, by the properties of expectation, for the sample we have $\mathbb{E}[\bar{x}]=p$. So here the sample mean $\bar{x}$ provides an estimate of the population parameter $p$. In statistics it is standard practice to denote an estimate of a population parameter by using a "hat", so here we it makes sense to denote the sample mean as $\hat{p}$.

(For the "xbar" problem the comparable notation would be $\bar{x}=\hat{\mu}$, as there $x$ is normal rather than Bernoulli.)

• Very informative! Makes me appreciate statistics more now then when I took my Business Stats class now that I've taken Discrete Mathematics, because I loved discrete and now seeing it in statistics just makes me happy. – John Dec 7 '16 at 22:56

Below one could be a handy tip. The image clearly distinguishes between sample mean and sample proportions.

Source info: UF Biostatistics Open learning textbook, Module 9, Sampling Distribution of the Sample Mean (in case link dies out in future)

• Without any additional comments this table may be very misleading. – Tim Aug 9 '18 at 13:36