Final answer:
The point estimator for the population mean is the sample mean, which equals the population mean if the sample size is sufficiently large according to the Central Limit Theorem. When the sample size increases, the sample mean approximates the population mean more closely following the law of large numbers.
Step-by-step explanation:
The point estimator for the population mean is the sample mean. It is used as an estimate of the unknown population mean (μ). According to the Central Limit Theorem, if we have a sufficiently large sample size, we can expect the distribution of the sample means to be approximately normal. The mean of the sample means will equal the population mean, and the standard error of the mean will be the population standard deviation divided by the square root of the sample size.
In the provided example where a population mean is 13, the sample mean of 12.8 with a standard deviation of two and a sample size of 20 indicates that the sampling distribution of the sample mean will be normal because we are assuming the underlying population is normal. Hence, the normal distribution should be used to perform a hypothesis test.
When we take repeated samples, we would expect that the sample mean would be a good estimate of the population mean. The law of large numbers supports this, stating that as the sample size increases, the sample mean tends to approach the population mean. Additionally, in confidence interval calculations, we might state with 95 percent confidence that the true population mean μ is within a certain range around the sample mean.