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 
. For a random sample 
the sample mean will then be denoted by 
. This applies directly to the "xbar" question. Here each 
is a temperature measurement, and the question asks about the sampling distribution of 
. (This arises when 
is computed many times over different samples, each of size 
).
For the "phat" question, the notation and logic is consistent with this, but the connection is a little more involved. In this case each 
will correspond to an individual student, who either invests (
) or does not (
). The probability that a student will invest would commonly be denoted by 
(
in this case). These conventions of 
and 
are standard for the case of a binary random variable.
Now imagine we do not know the value of 
, but wish to estimate it from a random sample of students 
. For a single student the expected value of 
is 
, denoted 
(see also here). Similarly, by the properties of expectation, for the sample we have 
. So here the sample mean 
provides an estimate of the population parameter 
. 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 
.
(For the "xbar" problem the comparable notation would be 
, as there 
is normal rather than Bernoulli.)
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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 . For a random sample the sample mean will then be denoted by . This applies directly to the "xbar" question. Here each is a temperature measurement, and the question asks about the sampling distribution of . (This arises when is computed many times over different samples, each of size ).
For the "phat" question, the notation and logic is consistent with this, but the connection is a little more involved. In this case each will correspond to an individual student, who either invests () or does not (). The probability that a student will invest would commonly be denoted by ( in this case). These conventions of and are standard for the case of a binary random variable.
Now imagine we do not know the value of , but wish to estimate it from a random sample of students . For a single student the expected value of is , denoted (see also here). Similarly, by the properties of expectation, for the sample we have . So here the sample mean provides an estimate of the population parameter . 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 .
(For the "xbar" problem the comparable notation would be , as there is normal rather than Bernoulli.)
Below one could be a handy tip. The image clearly distinguishes between sample mean and sample proportions.
Source
Source info: UF Biostatistics Open learning textbook, Module 9, Sampling Distribution of the Sample Mean (in case link dies out in future)