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Cumulative probability represents the total probability that a random variable is less than or equal to a certain value. When dealing with a normal distribution, we often refer to the cumulative distribution function (CDF). The CDF gives us the area under the curve of the distribution to the left of a particular value.
For example, if you want to know the probability that a standard normal random variable (Z) is less than or equal to 1.5, you would use the Z table to find the cumulative probability up to 1.5. This value tells you the likelihood that your variable falls within that range.
Understanding cumulative probabilities is essential when performing probability calculations, as it allows you to determine the portion of data you are interested in.

The Z-score represents the number of standard deviations a given data point is from the mean of the distribution. For the standard normal distribution, the mean is 0 and the standard deviation is 1.
The Z-score helps convert any normal distribution to the standard normal distribution, making probability calculations and comparisons possible.
To calculate a Z-score, you use the formula: \(Z = \frac{(X - μ)}{σ}\), where X is the value, μ is the mean, and σ is the standard deviation.
For example, if you are given a value of 20, a mean of 10, and a standard deviation of 5, the Z-score is \(Z = \frac{(20 - 10)}{5} = 2\).
This means the value of 20 is 2 standard deviations above the mean.

The standard normal distribution table, or Z-table, provides cumulative probabilities for Z-scores in a standard normal distribution. It typically shows the probability that a standard normal variable is less than or equal to a given Z-score.
When using the Z-table:

• Find the Z-score row corresponding to the integer part and the first decimal place of your Z-score.
• Find the column that matches the second decimal place of your Z-score.
• Locate the value at the intersection of your selected row and column.

For example, to find \(P(Z \leq 1.34)\), locate the row for 1.3 and the column for 0.04. The intersection gives you the cumulative probability.
The Z-table is a fundamental tool in statistics for finding cumulative probabilities quickly and accurately.

Probability calculations with the standard normal distribution involve using the Z-table to find cumulative probabilities and then performing basic arithmetic operations.

• To find \(\text{Pr}(-1.3 \leq Z \leq 0)\), you first find \(\text{Pr}(Z \leq 0) = 0.5\) and \(\text{Pr}(Z \leq -1.3) = 0.0968\). Then, subtract to get the desired probability: \(0.5 - 0.0968 = 0.4032\).
• To find \(\text{Pr}(0.25 \leq Z)\), find \(\text{Pr}(Z \leq 0.25) = 0.5987\), then subtract it from 1: \(1 - 0.5987 = 0.4013\).
• To find \(\text{Pr}(-1 \leq Z \leq 2.5)\), find \(\text{Pr}(Z \leq 2.5) = 0.9938\) and \(\text{Pr}(Z \leq -1) = 0.1587\), and then subtract: \(0.9938 - 0.1587 = 0.8351\).

These calculations help understand the likelihood of a random variable falling within specified ranges.