Problem 12 - Thinking Mathematically

Question

In Exercises 9-16, find the percentage of data items in a normal distribution that lie between \(z=2\) and \(z=3\).

Step-by-Step Solution

Answer

The percentage of data items that lie between z=2 and z=3 in the normal distribution is approximately 2.15%.

Step 1: Use the Z-table to find the corresponding value for z=2

Look up z=2 in the Z-table. The value you get represents the proportion of data that falls below z=2; in this case, the value is 0.9772.

Step 2: Use the Z-table to find the corresponding value for z=3

Then look up z=3 in the Z-table. The value you get represents the proportion of data that falls below z=3; in this case, the value is 0.9987.

Step 3: Find the proportion of data that lies between z=2 and z=3

Subtract the value for z=2 from the value for z=3, i.e., 0.9987 - 0.9772 = 0.0215.

Step 4: Convert the decimal to a percentage

To express your answer as a percentage, multiply your result by 100, i.e., 0.0215 * 100 = 2.15%.

Key Concepts

Z-table

The Z-table is an essential tool for anyone working with the normal distribution. Also known as the standard normal table, it provides cumulative probabilities associated with a standard normal curve. This tool allows us to find the probability that a standard normal random variable is less than or equal to a particular value, "z."

A Z-table typically shows the cumulative probability for values ranging from -3.49 to 3.49. When looking up a particular z-score, it's important to use the Z-table accurately:

• Locate the z-score on the table, which is often split into a tenths position on the left and the hundredths position on the top. This means first find the vertical value that matches the integer and first decimal number of your z-score.
• Then, intersect this with the horizontal value representing the second decimal of your z-score to find the cumulative probability value.

In this exercise, users found the cumulative probability for z-scores 2 and 3, obtaining values of 0.9772 and 0.9987, respectively. This means that around 97.72% and 99.87% of the data lies below these z-scores.

percentage calculation

Calculating percentages from probabilities helps us interpret the data more intuitively. Percentages make it easier to comprehend complex statistical information at a glance. To convert a probability to a percentage, simply multiply by 100. This step is a straightforward but essential part of data analysis.

Once you have the probabilities from the Z-table, as seen in the exercise, you can calculate the percentage of data that falls within a specified range. For the range between two z-scores, subtract the smaller cumulative probability from the larger one. This gives the proportion of data that specifically lies between these two values.

Let's look at an example:

• You have cumulative probabilities from the Z-table for z=2 and z=3, which are 0.9772 and 0.9987 respectively.
• Subtract these values (0.9987 - 0.9772) to find the probability that a data point falls between z=2 and z=3. This gives 0.0215.
• Finally, multiplying 0.0215 by 100 yields 2.15%, indicating 2.15% of data lies between these two z-scores.

standard normal curve

The standard normal curve is a bell-shaped curve that describes how data is distributed in a standard normal distribution. In this standardized form, the mean is 0, and the standard deviation is 1, which simplifies calculations and comparisons.

Visualizing data on the standard normal curve can aid in understanding the distribution of data points. The tails of the curve represent extreme values, while most of the data clusters around the mean, under the high middle of the curve. This symmetry is key for various statistical applications.

Within this curve, each z-score corresponds to a precise location. For the exercise, values such as z=2 or z=3 are specific points along this curve, helping us determine the proportion of data below or between certain points.

• The empirical rule applies here, suggesting that approximately 68% of the data falls within one standard deviation from the mean, about 95% within two, and 99.7% within three.
• This rule offers quick insights but always check the Z-table for accurate cumulative probabilities.

The standard normal curve serves as a fundamental concept when discussing probabilities and percentages in normally distributed data.