Question

Find the probability that \(z\) lies between \(z=-1.48\) and \(z=1.48\).

Short answer

Answer: The probability that a variable z lies between z = -1.48 and z = 1.48 is approximately 0.8612, or 86.12%.

Step-by-step solution

Identify the known variables

In this exercise, we are given the minimum and maximum z-scores: \(z_1 = -1.48\) and \(z_2 = 1.48\).

Find the area to the left of \(z_1\) and \(z_2\)

Use the standard normal distribution table (or an online calculator, such as a z-table calculator) to find the area to the left of each z-score. Look up each z-score in the table and read off the corresponding probability. For \(z_1 = -1.48\), the probability is approximate \(0.0694\). This means that approximately \(6.94\%\) of the data lies below this z-score. For \(z_2 = 1.48\), the probability is approximate \(0.9306\). This means that approximately \(93.06\%\) of the data lies below this z-score. (Note that these probabilities might slightly vary depending on the z-table you use).

Calculate the probability between \(z_1\) and \(z_2\)

To find the probability that \(z\) lies between \(z_1\) and \(z_2\), we subtract the area to the left of \(z_1\) from the area to the left of \(z_2\), as follows: \(P(z_1 \leq z \leq z_2) = P(z \leq z_2) - P(z \leq z_1) = 0.9306 - 0.0694 = 0.8612\)

Interpret the result

The probability that a variable \(z\) lies between \(z = -1.48\) and \(z = 1.48\) is \(0.8612\), which means that there is an \(86.12\%\) chance that a data point will fall between these z-scores.

Key concepts

Understanding Z-Scores

In the realm of statistics, the concept of a 'z-score' is fundamental. A z-score signifies how many standard deviations an element is from the mean of a distribution. It is calculated with the formula: \( z = \frac{(X - \mu)}{\sigma} \), where \( X \) is the value being standardized, \( \mu \) is the mean of the distribution, and \( \sigma \) is the standard deviation.

When we talk about z-scores, we are essentially standardizing individual data points, which makes different data sets comparable. A z-score of 0 represents the mean of the distribution, while a positive z-score indicates a value above the mean, and a negative z-score represents a value below the mean. This standardization is crucial for finding probabilities, as it allows us to refer to a standard normal distribution table, regardless of the original distribution of the data.

Why Z-Scores Matter

Understanding z-scores is pivotal in various statistical analyses, like hypothesis testing and regression analysis. They help to quantify the unusualness of a data point within the context of a given data set, providing a clear and objective measure.

The Standard Normal Distribution

The standard normal distribution, also known as the z-distribution, is a special case of the normal distribution. It is a bell-shaped curve that is symmetric about the mean, located at a z-score of 0, and has a standard deviation of 1. This aspect of symmetry implies that half of the values lie below the mean and half above it.

When dealing with probabilities, we make the assumption that the underlying data follows a normal distribution, allowing us to harness the standard normal model to make predictions and conclusions about our data. Crucially, any normal distribution can be transformed into the standard normal distribution via the use of z-scores, which aids in simplifying complexity across different data sets with varying scales of measurement.

Importance of Standardization

Standardizing values enable comparability across data sets which may originally have different scales or units. This is why converting raw scores to z-scores is such a key step in probability calculation and other statistical methodologies.

Navigating the Normal Distribution Table

The normal distribution table, often known as the z-table, is an essential tool used in statistics for determining the probability that a statistic is observed below, above, or between certain values in a normal distribution. It represents the area under the curve to the left of a given z-score.

To use this table, you locate the z-score on the grid, and that intersection gives you the cumulative probability. For example, if you wanted to know the percentage of data falling below a z-score of 1.48, you'd find the corresponding value on the z-table, which would read as approximately 0.9306 or 93.06%. Meaning, there's a 93.06% chance that a data point chosen at random will have a z-score less than 1.48.

• Finding Probabilities: The table enables you to find the likelihood of occurrences within a normal distribution framework.
• Subtract to Find Between: By subtracting the probability of one z-score from another, you can determine the probability that a statistic falls between two values.

It is, however, important to note that precision depends on the z-table version used. As students, being adept at reading a z-table is crucial for exercises involving probability and statistics.