A random sample of n = 15 items are selected

Introduction:
The sample size considered here is, n = 15.
Explanation:
It is of interest to construct a confidence interval for the population mean measurement.
Now, an unbiased point estimate for the population mean is the sample mean. As a result, the confidence interval is constructed using the sample mean, its standard error, and a confidence level. The construction of the confidence interval for the population mean requires the basic assumption of at least an approximate normal distribution of the variable of interest.
The sampling distribution of the sample mean, ${\displaystyle \bar{X}}$, based on a sample of size n, taken from a population with expectation ${\displaystyle \mu }$ and standard deviation ${\displaystyle \sigma }$, has expectation ${\displaystyle {\mu }_{\bar{X}}=\mu }$ and standard deviation ${\sigma }_{\bar{X}}=\frac{\sigma }{\sqrt{n}}$
If the sample size is large ${\displaystyle (n\ge 30)}$, or the population distribution is normal, then by the central limit theorem, the sampling distribution of the sample mean is normal, with parameters ${\displaystyle {\mu }_{\bar{X}}}$ and ${\displaystyle {\sigma }_{\bar{X}}}$.
In this case, no information is available regarding the distribution of the measurements. As a result, it cannot be said that the measurements are normally distributed.
Further, the sample size of n = 15 is not large. Thus, it is not possible to assume even an approximate normal distribution.
Thus, the requirements for constructing a confidence interval for the population mean are not satisfied.