Answer :
Using 68-95-99.7 rule Pr(z < -2) is also approximately 2.5%.
The 68-95-99.7 rule, also known as the empirical rule or the three-sigma rule, is a rule of thumb that applies to a normal distribution.
It states that approximately 68% of the data falls within one standard deviation of the mean, about 95% falls within two standard deviations, and around 99.7% falls within three standard deviations.
To determine Pr(z < -2) using the 68-95-99.7 rule, we need to understand the concept of the standard normal distribution.
In a standard normal distribution, the mean is 0 and the standard deviation is 1.
In this case, we are interested in finding the probability of a z-score less than -2. A z-score represents the number of standard deviations a data point is away from the mean.
It helps us standardize and compare values from different normal distributions.
To find Pr(z < -2), we can use a standard normal distribution table or a calculator.
When we look up the z-score of -2 in the table, we find that the corresponding probability is approximately 0.0228.
Therefore, Pr(z < -2) is approximately 0.0228, which means that there is a 2.28% chance of observing a value less than -2 standard deviations from the mean in a standard normal distribution.
It's important to note that the 68-95-99.7 rule is an approximation and the exact probabilities may differ slightly, but it provides a good estimate for understanding the distribution of data in a normal distribution.
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