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The Central Limit Theorem (CLT) is a fundamental principle in statistics. It tells us that the sampling distribution of the sample mean will become approximately normal as the sample size increases, even if the original population distribution is not normal. This holds true especially if the sample size is greater than or equal to 30.
However, in practice, if the population itself is already normally distributed, the sample mean's distribution is also normal, regardless of the sample size.
This means we didn’t even need a sample size as large as 30 in this exercise. With our sample size of 16 and given the population’s normal distribution, we can safely assume that our sample mean distribution is normal.
This allows us to move forward with hypothesis testing using the normal distribution properties.

The Z-score helps us understand how far the sample mean is from the hypothesized population mean, under the null hypothesis, in units of standard deviation.
To calculate it in this context, we use the formula:

• \[ z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}} \]

Where:

• \(\bar{x}\) is the sample mean,
• \(\mu_0\) is the population mean under the null hypothesis,
• \(\sigma\) is the population standard deviation,
• \(n\) is the sample size.

In our exercise, plugging in the numbers gives us a \(z\)-score of -2.4. This negative \(z\)-score indicates that our sample mean is 2.4 standard deviations below the hypothesized population mean.
This metric serves as a crucial step to determining how likely it is to observe a sample mean as extreme as ours, assuming the null hypothesis is true.

The \(P\)-value represents the probability of obtaining a result at least as extreme as the one observed, given that the null hypothesis is true.
In our case, once we calculated the \(z\)-score, we could look up a \(P\)-value associated with \(z = -2.4\). This gave us a \(P\)-value of approximately 0.0082.
This small \(P\)-value indicates that there is only a 0.82% chance of observing a sample mean as extreme as 4.5 if the true population mean is actually 6.3.
Such a low \(P\)-value suggests that the sample mean we have is unusually low or rare, assuming the null hypothesis is correct. Hence, it contributes significantly to the decision-making in hypothesis testing.

The level of significance, denoted by \(\alpha\), represents a threshold for determining whether the \(P\)-value is small enough to reject the null hypothesis.
In most hypothesis tests, common levels of significance are 0.05, 0.01, or 0.10. This basically means there is a 5%, 1%, or 10% risk, respectively, of rejecting the null hypothesis incorrectly.
In our exercise, the level of significance is set at 0.01. Comparing this with our \(P\)-value of 0.0082, since \(0.0082 < 0.01\), we have sufficient evidence to reject the null hypothesis at the 1% risk level.
This implies that the observed sample mean of 4.5 provides strong evidence against the null hypothesis that the population mean is 6.3, affirming that this result is highly unlikely due to random chance.