Problem 31 - Algebra 2

Question

A set of 260 data values is normally distributed with a mean of 50 and a standard deviation of \(5.5 .(\text { (lesson } 12-7)\) What is the probability that a data value selected at random is greater than 39\(?\)

Step-by-Step Solution

Answer

The probability that a data value is greater than 39 is approximately 0.9772.

Step 1: Understanding the Problem

The problem provides a normal distribution with a mean (\(\mu\)) of 50 and a standard deviation (\(\sigma\)) of 5.5. We want to find the probability that a randomly selected data value is greater than 39.

Step 2: Find the Z-score

The Z-score helps us find the probability related to a specific value in a normal distribution. It is calculated using the formula: \[ Z = \frac{X - \mu}{\sigma} \]where \(X\) is the value of interest (39), \(\mu\) is the mean (50), and \(\sigma\) is the standard deviation (5.5). Substituting these values in gives: \[ Z = \frac{39 - 50}{5.5} = \frac{-11}{5.5} \approx -2.00 \]

Step 3: Use Standard Normal Distribution Table

Consult a standard normal distribution table (Z-table) to find the probability of a Z-score. A Z-score of -2.00 corresponds to a probability of approximately 0.0228, indicating that the probability that a value is less than 39 is 0.0228.

Step 4: Calculate Probability Greater Than 39

Since the probability of a data value being less than 39 is 0.0228, the probability of it being greater than 39 is 1 minus this value:\[ P(X > 39) = 1 - P(X < 39) = 1 - 0.0228 = 0.9772 \]

Key Concepts

Probability

Probability is a measure of how likely an event is to occur. In the context of a normal distribution, probability helps us understand the likelihood of a certain data point falling below, above, or between specific values.
A normal distribution is symmetric and follows the familiar bell curve shape. The total area under the normal distribution curve equals 1, which corresponds to a probability of 100%. This means that all events are accounted for under the curve.
When we talk about the probability of a data point being greater than a specific value, we are interested in the area of the curve that lies to the right of the value on the horizontal axis. This area represents the proportion of all data points that are greater than the specified value.

• Probability of any single point is technically 0 in continuous distributions like normal, but we use the area under the curve to represent the likelihood of a range of values.
• Finding probabilities in normal distribution often involves the calculation using a Z-score, which helps translate the problem into a standard normal distribution context.

Z-score

The Z-score is a powerful tool in statistics that helps us understand how far away a particular data point is from the mean of a distribution. It's given in terms of the number of standard deviations a value lies from the mean.
The formula to calculate the Z-score is: \[ Z = \frac{X - \mu}{\sigma} \]where

• \(X\) is the data value we are looking at (in this case, 39),
• \(\mu\) is the mean of the distribution (here, 50),
• \(\sigma\) is the standard deviation (5.5 in this problem).

By transforming a data value to a Z-score, we can easily find its probability using a Z-table. The Z-score we calculated, -2.00, tells us that the value 39 is 2 standard deviations below the mean. On a standard normal distribution curve, this corresponds to a specific probability.

Standard Deviation

Standard deviation is a measure of the amount of variability or spread in a set of data values. In a normal distribution, it helps define the shape of the curve. The larger the standard deviation, the wider the curve, indicating more spread out data points.
A smaller standard deviation indicates that the data points are closer to the mean. In the context of this problem, a standard deviation of 5.5 implies that on average, data values deviate 5.5 units from the mean.
Standard deviation allows us to understand the distribution of data and is integral to calculating Z-scores. Knowing how data is dispersed around the mean helps us better grasp the distribution and predict probabilities associated with specific ranges of values.

• One standard deviation from the mean covers approximately 68% of the data in a normal distribution.
• Two standard deviations cover about 95%, and three standard deviations cover about 99.7%.

Mean

The mean, often referred to as the average, is the central value of a data set. In a normal distribution, the mean is located at the highest point of the curve, dividing the distribution into two equal halves. It gives us a sense of the central tendency of the data.
In this exercise, the mean is given as 50, which means that the average of all the data points in the set is 50. The mean acts as a reference point when calculating Z-scores and understanding how the data is distributed.
Since the normal distribution is symmetric about the mean, knowing the mean allows us to predict the likelihood and probability of data points falling to the right or left of this central point.

• The mean is true only for symmetric, bell-curved (normal) distributions and crucial for calculating probabilities using standard normal tables.
• It's a key parameter in defining the normal curve alongside standard deviation.