91Ó°ÊÓ

A z-table, also known as a standard normal distribution table, is a chart that helps in finding the probability that a statistic is less than a given z-value. The z-table displays cumulative probabilities, which represent the area under the standard normal curve up to a certain z-value. When you look at a z-table, the numbers on the leftmost column represent the whole number and the first decimal place of a z-value. Meanwhile, the numbers on the top row correspond to the second decimal place of the z-value. To use the z-table effectively:

• Identify the z-value you're interested in. In our example, it's \( z = -0.75 \).
• Find the row with \( -0.7 \) and locate the column under \( 0.05 \).
• The intersection gives the cumulative probability for \( z = -0.75 \).

It's important to understand that z-tables typically reflect cumulative probabilities from the mean of the standard normal distribution, which is zero, up to the z-value of interest.

Cumulative probability is a measure used in statistics to find the probability that a random variable falls within a certain range of values. When dealing with the standard normal distribution, cumulative probability specifically refers to the probability that a z-score in question is less than a particular value. In other words, for a given z-value, cumulative probability tells us how much of the distribution lies to the left of that score.

In our problem with \( z = -0.75 \), the cumulative probability shows the likelihood of finding a value from the distribution that is equal to or less than \( -0.75 \). Calculated by integrating under the curve up to the specific z-score, cumulative probability allows us to determine how extreme a value is relative to the whole data set. We discovered that approximately 22.66% of the dataset is located to the left of \( z = -0.75 \). This gives us valuable insight into the position and rarity of the z-score within the distribution.

The z-value, also known as a z-score, is a statistical measure that describes a value's position in terms of standard deviations from the mean of the distribution. If you know the mean and standard deviation of the data, you can calculate the z-value with the formula:\[ z = \frac{{X - \mu}}{{\sigma}} \]where \( X \) is the value in the dataset, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.

The z-value helps compare different data points across different normal distributions by converting to standardized units. Remember that negative z-values indicate points below the mean, while positive ones are above it. Using a z-value like \( z = -0.75 \), enables the determination of cumulative probabilities via z-tables, as it standardizes the discussion regardless of the original scale of measurement. This function is particularly useful in comparing scores across different scales or distributions.