Problem 16 - Thinking Mathematically

Question

In Exercises 9-16, find the percentage of data items in a normal distribution that lie between \(z=-2.2\) and \(z=-0.3\).

Step-by-Step Solution

Answer

Therefore, approximately 36.82% of the data items in the distribution lie between \(z=-2.2\) and \(z=-0.3\).

Step 1: Locate the Z-scores

Looking at the given problem, we are trying to find the percentage of data items between \(z = -2.2\) and \(z = -0.3\). Start by locating these two z-scores in the z-table. Note that the z-table gives probabilities from the mean (z=0) up to the z-score.

Step 2: Find the area for each Z-score

Find the area to the left of each z-score by reading the value from the z-table corresponding to each score. This represents the proportion of the data that is below each given z-score. If you can't find the exact z-score, round to the nearest available value. For \(z=-2.2\), the area is approximately 0.0139, and for \(z=-0.3\), it's about 0.3821.

Step 3: Calculate the area between the two Z-scores

To find the area (percentage of data items) between the two z-scores, subtract the area of the lower z-score from the area of the higher z-score. In this case, subtract the area of \(z=-2.2\) from the area of \(z=-0.3\): 0.3821 - 0.0139 = 0.3682 or 36.82%

Key Concepts

z-score

Understanding the z-score is essential for analyzing data in a normal distribution. A z-score, also known as a standard score, provides an idea of how far away a particular point is from the mean, in terms of standard deviations.

For example, if we have a z-score of -2.2, it tells us that this point is 2.2 standard deviations below the mean. Conversely, a z-score of -0.3 indicates a point that is only 0.3 standard deviations below the mean. Calculating z-scores allows us to compare data from different normal distributions by standardizing the scores.

percentile rank

The percentile rank of a score is the percentage of scores in its frequency distribution that are equal to or lower than it. It is a way of ranking data points based on their relative position within a distribution.

A percentile rank is often associated with its corresponding z-score in a normal distribution. For instance, a score with a z-score of -0.3 might be at the 38.79th percentile, meaning approximately 38.79% of the data would fall below or at this score. Percentile ranks help us understand how an individual score compares to the rest of the dataset.

probability in normal distribution

In the realm of statistics, discovering probabilities in a normal distribution is a common task. Probability in this context refers to the likelihood of a particular outcome occurring within the distribution.

By using the z-scores and the properties of the normal distribution, we can determine these probabilities. As showcased in the exercise, to find the likelihood of a data point falling between two z-scores, we calculate the area under the curve of the normal distribution between those scores. This area translates directly to the probability of finding a data point within that range.

z-table

A z-table, also known as the standard normal table, is a reference chart that we use to find probabilities associated with specific z-scores in a standard normal distribution. This table displays the cumulative probability from the mean to any z-score.

When working with a z-table, note that it shows you the area to the left (below) the designated z-score. Hence, to find the percentage of data between two z-scores such as -2.2 and -0.3, you would look up both values and subtract the smaller area from the larger one, as demonstrated in our step-by-step solution. This method will yield the probability of a data item falling between these two z-scores.