Quick questions on Distribution of the sample mean and the central limit theorem (QCE Specialist Mathematics Unit 4)

If you take a random sample of size $n$ from a population and compute its mean $\bar{X}$, a different sample gives a different mean. So $\bar{X}$ is a random variable. Its distribution is called the sampling distribution of the mean, and it has its own mean and standard deviation.

The central limit theorem states that for a sufficiently large sample size $n$, the distribution of the sample mean $\bar{X}$ is approximately normal,

To compute a probability for $\bar{X}$, standardise using the standard error:

Because the standard error is $\dfrac{\sigma}{\sqrt{n}}$, quadrupling the sample size halves the spread of $\bar{X}$. This is why larger samples produce tighter estimates and narrower confidence intervals: the sampling distribution concentrates around $\mu$.

A favourite extended-response task gives a probability statement about $\bar{X}$ and asks for the sample size. The method reverses standardising: convert the probability to a critical $z$-value, then solve $\dfrac{\text{(value} - \mu)}{\sigma/\sqrt{n}} = z$ for $n$. Because $n$ must be a whole number of observations, round the result, and state explicitly that a sample size is an integer. Reading the correct $z$ from the stated tail probability (lower tail, upper tail, or central region) is where care is needed, and a sketch of the normal curve with the area shaded prevents sign errors.