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In statistics, a z-score, also known as a standard score, is a way to describe the position of a raw score in relation to the mean of a group of scores. It is a measure of how many standard deviations an element is from the mean. For this reason, it's a powerful tool for comparing results across different data sets.
The formula for calculating a z-score is:

• \( z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}} \)

Where:

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

In our exercise, we calculate a z-score to assess whether the sample mean significantly differs from the population mean. A z-score of \(-2\) indicates the sample mean is 2 standard deviations away from the population mean, suggesting a potential difference.

The t-test is a statistical test that is used to compare the means of two groups. It's particularly useful when the population standard deviation is unknown, which often happens in real-life scenarios. Instead of using the population standard deviation, it uses the sample standard deviation and is applied using the t-distribution, which accounts for the smaller sample size by having 'fatter' tails than the normal distribution.
The formula for calculating the t-score is:

• \( t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}} \)

Where:

• \( \bar{x} \) is the sample mean,
• \( \mu \) is the population mean,
• \( s \) is the sample standard deviation,
• \( n \) is the sample size.

In our battery life example, a t-score of \(-2\) was calculated. However, the evidence wasn't strong enough to reject the null hypothesis when the sample standard deviation was used. This showcases the importance of considering sample size and variance when drawing conclusions from data.

Standard deviation is a measure that indicates the amount of variation or dispersion in a set of values. Simply put, it tells us how much the individual data points differ from the mean of the data set. A small standard deviation means the data points are close to the mean, while a large one indicates they are spread out over a wider range of values.
In statistical testing, like in our battery life test, standard deviation plays a crucial role. It allows us to understand how much our sample results can vary. In cases where the population standard deviation is known, we use it directly in z-score calculations. When unknown, we rely on the sample standard deviation for t-tests. This transformation ensures our tests accommodate variation in data, providing more accurate results.