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The Z-score is a fundamental concept when working with normal distributions. It is a measure of how many standard deviations a data point, such as a battery life in our case, is from the mean value. In simple words, the Z-score tells us where a specific value stands in relation to the average. The formula for calculating a Z-score is:\[z = \frac{X - \mu}{\sigma}\]where:

• \( X \) is the data point for which we want to find the Z-score,
• \( \mu \) is the mean of the distribution, and
• \( \sigma \) is the standard deviation of the distribution.

In our example, we calculated a Z-score for a battery life of 240 minutes. By substituting into the formula, we obtained a Z-score of \(-0.4\), indicating that the battery life of 240 minutes is 0.4 standard deviations below the mean of 260 minutes. Understanding Z-scores is crucial for finding probabilities and identifying where data lies relative to a normal distribution.

Probability helps us determine how likely an event is to occur within a particular distribution. For a normally distributed dataset like battery life, we can use the Z-score to find the probability of certain outcomes.
Once we have a Z-score, we can refer to a Z-table, a tool that tells us the probability of a value falling below a given Z-score. This is known as a cumulative probability. In the battery example, a Z-score of \(-0.4\) had a cumulative probability of 0.3446. That means there's a 34.46% chance that a battery won't last up to 240 minutes.

• To find the probability of a battery lasting more than a certain time, we subtract the cumulative probability from 1. This gives us the probability that a battery will exceed that time.
• In our instance, the probability that a battery lasts more than 240 minutes is 0.6554, or 65.54%.

Understanding probability in the context of normal distribution helps us make predictions and decisions based on statistical data.

Quartiles are values that divide a data set into four equal parts, helping us understand the data's distribution. In statistics, the first quartile (Q1) and the third quartile (Q3) show where 25% and 75% of the data points lie, respectively.
Calculating quartiles involves finding specific Z-scores and converting them into actual values from the data set. For a normal distribution:

• The 25th percentile Z-score is approximately \(-0.674\), and for the laptop battery, this corresponds to a life of about 226.3 minutes.
• The 75th percentile Z-score is approximately 0.674, translating to a battery life of around 293.7 minutes.

These values help describe the spread of the dataset. The quartiles tell us that 25% of batteries will last less than 226.3 minutes, and 75% will last more than this but less than 293.7 minutes. Analyzing quartiles can help manufacturers and users understand battery performance better.