Probability that population mean is less than $3$ given sample $\text{{1, 3, 3, 5, 9}}$

Question

So I was wondering if you were given a sample within a population of unknown size, how you would determine the chance that the population's mean is less than $3$. The sample is $\text{{1, 3, 3, 5, 9}}$. I know about the standard deviation ($3.03$) and mean of the sample ($4.2$), that the population is normally distributed, and the expected value of the population's mean ($4.2$).

My approach to the problem was to assume population size $5$, find its standard deviation, and find the probability that the population mean is less than $3$. Then repeat with population size $6, 7, 8, 9...$ and find the average of the probabilities since each population size is equally likely. This is how the math looks like: $$P(\mu \leq 3) = \lim_{t \to \infty} \frac {\sum ^{t}_{i = 5} P_i(\mu_i \leq 3)}{t-5}$$ where $P_i$ is the probability that the mean is less than $3$ for population size $i$. Unfortunately, I cannot find this function without knowing the standard deviation of the population. How do I find the standard deviation knowing this information (in $1$st paragraph)?

Answer 1

This is the kind of question that Bayesian statistics answers.

The traditional way of doing statistics that gets covered early on (AP statistics, for instance) is called frequentist statistics. It is not that frequentist statistics gets it wrong or is worthless, but frequentist statistics cannot and should not answer this question.

Frequentist statistics answers how likely it is to get a mean of at least 4.2, under the assumption that the population has a mean of 3, which is somewhat of the reverse of what you seek. (This is just about the definition of a p-value.)

Bayesian statistics, however, combines your observations with some prior beliefs; this is the prior distribution. If you believe, before collecting data, that you have a population that’s $N(3,1)$, your five observations do not change your belief very much. The combination of prior beliefs and data form the posterior distribution of $\mu$, which would allow you to calculate the probability that $\mu<3$.

However, you must make some assumption for that prior belief. Without that, the question cannot be answered.

Answer 2

Are you familiar with the difference between the Frequentist/Bayesian approaches of statistical inference? The reason I'm asking is because, asking:

(1) What is the chance the mean of a population is less than x?

Is a valid question within the Bayesian framework and not a valid question within the frequentist framework.

In the Frequentist framework we assume the parameter we wish to infer about (in your case, the population mean) has a fixed unknown value. In this school of thought, it doesn't make any sense to ask what's the probability that the population mean is less than 3 as the mean is either 3 or not - the mean isn't a random variable. We can however test the hypothesis that the population mean is less than three. The difference is, that now we are asking:

(2) Is the population mean less than 3?

Notice that this question doesn't involve asking about the probability of the values the mean can take. In order to answer (2) we typically use hypothesis testing. PennState and StatTrek have some guides on hypothesis testing. For your problem, you are should look for the one sided test of a mean.

In the Bayesian paradigm the unknown parameter is treated as a random variable, this allows us to ask questions such as (1). The randomness is incorporated into the unknown parameter by giving it a prior distribution. This prior distribution should reflect our beliefs on the possible values that the parameter can take, possibly due to past experience. You might for example know that $\mu\sim N(0,1)$. The Bayesian inference is based on the posterior distribution of the parameter, which is the distribution of the parameter given the observed data. I think it's best to understand frequentist inference before approaching the Bayesian solution. If your'e looking into a learning more, this is a very good book and concise book on statistical theory (Abramovich was my lecturer for statistical theory).