91Ó°ÊÓ

The Z-score is a fundamental concept when dealing with the normal distribution. It provides a way to determine how far away a specific data point is from the mean, in terms of standard deviations. This statistical measure helps us understand where a data point falls within the distribution.
The formula for the Z-score is given by:\[ Z = \frac{X - \mu}{\sigma} \]Where:

• \(X\) is the value of interest.
• \(\mu\) is the mean of the distribution.
• \(\sigma\) is the standard deviation.

In the context of our exercise, calculating the Z-score for a bottle containing 295 ml helps identify how uncommon or far from the average it is. A Z-score of -1, for instance, shows that the bottle is 1 standard deviation below the mean.

Once the Z-score is determined, the next step is to find the probability associated with this Z-score. Probabilities in a normal distribution can be found using a Z-table or a normal distribution calculator. These tools provide the probability that a value lies below a specific Z-score.
For example, a Z-score of -1 corresponds to a probability of about 0.1587. This means there is a 15.87% chance that a randomly selected cola bottle contains less than 295 ml. Similarly, for a Z-score of -2.45, the probability is roughly 0.0071, indicating just a 0.71% chance that the average content of six bottles falls below 295 ml.
Understanding how to navigate and interpret Z-tables is crucial for these calculations, allowing predictions about how often certain values occur within your data set.

Sampling distribution is essential when examining the averages from multiple samples. It describes the distribution of a statistic, such as the sample mean, over numerous samples taken from the same population.
In our exercise, when multiple bottles are taken as a group, the mean content of these bottles becomes the focus, not just individual bottles. This collective distribution is also normal provided the underlying population distribution is normal (or close to it).
The mean of the sampling distribution is the same as the population mean, while the standard deviation, also known as the standard error, is calculated by dividing the population standard deviation by the square root of the sample size \( n \):\[ \sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}} \]For six bottles, this would adjust the standard deviation, helping understand how the average across these bottles deviates from the mean.

The Central Limit Theorem (CLT) is a cornerstone of probability theory. It states that the distribution of sample means approaches a normal distribution as the sample size grows, regardless of the original distribution's shape, provided the samples are independent and identically distributed.
This theorem underlies the trustworthiness of cooking modern-day statistics, allowing statisticians to make inferences. Even when the population distribution is not normal, the sample mean distribution will approximate a normal distribution if the sample is sufficiently large (usually \(n \geq 30\)).
In our bottling example, the sample size was just six, but because the original distribution is normal, the sampling distribution is also normal. This makes it possible to apply the concept of normal distribution, making calculations for probabilities straightforward and reliable, even for small sample sizes.