Standard Normal Distribution: Videos & Practice Problems

In the study of statistics, understanding continuous random variables and the standard normal distribution is crucial. The standard normal distribution is represented by a bell curve, characterized by its symmetry around the mean, which is zero. The z-score is a key concept here, as it quantifies how many standard deviations a data point is from the mean. To find probabilities associated with specific z-scores, we utilize a z-table, which provides cumulative probabilities for the standard normal distribution.

When tasked with finding the probability that a z-score is less than a given value, the process begins with sketching the normal curve and identifying the area of interest. For example, if we want to find the probability that z is less than -0.64, we would shade the area to the left of this z-score on the graph. Using the z-table, we locate the row for -0.6 and the column for 0.04, which intersect at the value 0.2611. This value represents the area under the curve to the left of -0.64, indicating that the probability of z being less than -0.64 is 0.2611.

Conversely, if we need to find the probability that z is greater than a certain value, such as 2.27, we again start by sketching the normal curve. This time, we are interested in the area to the right of the z-score. After locating 2.27 in the z-table, we find the cumulative probability of 0.9884, which represents the area to the left of 2.27. To find the area to the right, we subtract this value from 1, yielding a probability of 0.0116. This indicates that the probability of z being greater than 2.27 is 0.0116.

In summary, the z-table is an essential tool for determining probabilities associated with z-scores in the standard normal distribution. By visualizing the problem with graphs and carefully interpreting the values from the z-table, one can accurately find the probabilities of various outcomes in statistical analysis.

In statistics, understanding how to calculate probabilities using z-scores is essential, especially when determining the area between two z-scores. This process involves utilizing a z-table, which provides the cumulative probabilities associated with standard normal distribution values.

To find the probability between two z-scores, such as -0.64 and 2.27, you first need to identify the cumulative probabilities for each z-score from the z-table. For instance, the cumulative probability for z = -0.64 is 0.2611, while for z = 2.27, it is 0.9884. These values represent the area to the left of each z-score on the standard normal distribution curve.

To visualize this, you can sketch a standard normal distribution curve, marking the two z-scores. The area of interest is the region between these two points. To calculate the probability of this area, you subtract the cumulative probability of the smaller z-score from that of the larger z-score. This can be expressed mathematically as:

Probability (between z1 and z2) = P(Z < z2) - P(Z < z1)

In our example, this translates to:

Probability (between -0.64 and 2.27) = 0.9884 - 0.2611 = 0.7273

This result, 0.7273, represents the probability that a randomly selected value from a standard normal distribution falls between the z-scores of -0.64 and 2.27. This method is a straightforward and effective way to find probabilities between two z-scores, reinforcing the importance of understanding cumulative probabilities in statistical analysis.

Understanding how to find z scores from given probabilities under the standard normal curve is essential in statistics. This process often involves using a z table, which provides cumulative probabilities associated with z scores. When given a probability, the first step is to sketch the area under the normal curve to visualize the problem. This helps in determining whether the area corresponds to the left or right of a z score.

For instance, if you are given a probability of 0.8051, this represents the area to the left of a certain z score. Since the total area under the curve is 1, you can infer that a probability close to 0.5 indicates a z score near 0, while a probability close to 1 suggests a z score that is positive and further to the right. In this case, you would look up the value 0.8051 directly in the z table, which reveals that the corresponding z score is approximately 0.86.

Conversely, if you are given a probability that represents an area to the right of a z score, such as 0.67, you must first convert this to the area to the left. This is done by subtracting the given probability from 1, resulting in an area of 0.33. You would then look up this value in the z table, which typically contains cumulative probabilities from the left. Finding 0.33 in the z table indicates a z score of approximately -0.44, confirming that the z score is slightly negative, consistent with the area being more than half of the curve to the right.

In summary, when working with areas to the left of a z score, you can directly use the given probability in the z table. However, for areas to the right, always subtract the given probability from 1 to find the corresponding area to the left before consulting the z table. This systematic approach ensures accurate determination of z scores from probabilities.