91Ó°ÊÓ

In statistics, a standard normal variable is a crucial concept. It allows us to compare different normal distributions. The standard normal variable, often denoted as Z, represents a normal distribution with a mean of 0 and a standard deviation of 1. When we transform a normal variable X into a standard normal variable Z, we can use standard tables to find probabilities more easily.

Understanding the concept of standard normal variables is key to solving many types of problems in probability and statistics.
Transforming the variable follows this formula:

\[ Z = \frac{ (X - \mu) } {\sigma} \]
Here:

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

This conversion is crucial, as it allows us to use the standard normal distribution tables to find the required probabilities once these conversions are made.

The Z-score is a way of determining how many standard deviations a data point is from the mean. Calculating the Z-score is simple and essential for probability computations involving normal distributions.

To calculate the Z-score for a value X in a normal distribution with any mean \(\mu\) and standard deviation \(\sigma\), use the formula:

\[ Z = \frac{ (X - \mu) }{ \sigma } \]
In the given exercise, we're asked to find the probability \(P(X > 35)\). Here, X=35, \(\mu = 50\), and \(\sigma = 7\). Substituting these values into our Z-score formula gives us:

\[ Z = \frac{ (35 - 50) }{ 7 } = \frac{ -15 }{ 7 } \approx -2.14 \]
This Z-score tells us that 35 is approximately 2.14 standard deviations below the mean of 50 in this distribution.

Once the Z-score is determined, the next step is to find the probability associated with this Z-score using standard normal distribution tables.

These tables provide the area under the curve to the left of a given Z-score. For Z = -2.14, the table tells us that the area to the left of this Z-score is approximately 0.0162.

However, the problem requires us to find the probability that X is greater than 35, which is the area to the right of the Z-score. To compute this, we subtract the area to the left from 1:

\[ P(X > 35) = 1 - 0.0162 = 0.9838 \]
This means that there is approximately a 98.38% chance that a randomly selected value from this distribution is greater than 35.

The normal distribution curve, or bell curve, is key to understanding probabilities in a normal distribution. The total area under the curve represents all possible outcomes, and this area equals 1.

The area under the curve between any two points represents the probability of the variable falling within that range. For instance, in our exercise, the area to the right of the Z-score of -2.14 covered the probability \(P(X > 35)\). Visualizing this curve and shading the appropriate area helps in understanding the concept more effectively.

By sketching a normal distribution curve with a mean (\(\mu\)) of 50 and marking the value 35, then shading the area to the right, you can visually interpret the probability \(P(X > 35)\). This shaded area under the curve helps illustrate the concept of probability in a normally distributed variable.