Problem 44 - Algebra 2

Question

A set of 400 test scores is normally distributed with a mean of 75 and a standard deviation of 8 . What is the probability that a randomly-selected score is less than 67\(?\)

Step-by-Step Solution

Answer

15.87% probability that a score is less than 67.

Step 1: Identify the parameters

We start by recognizing the mean (\(\mu\)) and standard deviation (\(\sigma\)) of the normal distribution. Here, \(\mu = 75\) and \(\sigma = 8\). We are asked to find the probability that a random score is less than 67.

Step 2: Calculate the Z-score

To find the probability, we first calculate the Z-score for a score of 67 using the formula: \[ Z = \frac{X - \mu}{\sigma} \]Substituting the values, \(X = 67\), \(\mu = 75\), and \(\sigma = 8\), we get:\[ Z = \frac{67 - 75}{8} = \frac{-8}{8} = -1 \].

Step 3: Use the Z-table

With a Z-score of -1, we refer to the standard normal distribution Z-table to find the corresponding probability. The Z-table gives us the cumulative probability from the left. For a Z-score of -1, the cumulative probability is approximately 0.1587.

Step 4: Interpret the result

The cumulative probability of 0.1587 indicates that the likelihood of randomly selecting a score that is less than 67 is 15.87%. This means that about 15.87% of all scores fall below 67.

Key Concepts

Z-score

A Z-score is a way to describe a data point's position in relation to the mean of a dataset. It essentially tells us how far a given value is from the mean, measured in units of the standard deviation. Using the formula \( Z = \frac{X - \mu}{\sigma} \), where \( X \) is the value of interest, \( \mu \) is the mean, and \( \sigma \) is the standard deviation, we can calculate the Z-score for any point.
For example, if we're looking at a score of 67 from a dataset with a mean of 75 and a standard deviation of 8, the Z-score is calculated as \( \frac{67 - 75}{8} = -1 \). This negative value indicates that the score is one standard deviation below the mean.

• If a Z-score is zero, the data point is exactly at the mean.
• A positive Z-score means the data point is above the mean.
• A negative Z-score means it is below the mean.

Understanding Z-scores helps us compare different data points within the same dataset or across different datasets.

Cumulative Probability

Cumulative probability refers to the probability that a random variable is less than or equal to a certain value. In the context of normal distribution, we often use cumulative probability to determine the likelihood of a variable falling below a threshold.
When we calculated a Z-score of \(-1\), we used a Z-table to find the cumulative probability. This Z-table gives us the area under the standard normal distribution curve to the left of the Z-score. For \( Z = -1 \), this probability is approximately 0.1587, which means 15.87% of the values lie below this threshold.
Cumulative probability accumulates the probabilities of all outcomes up to and including the target value.

• It is useful for determining the proportion of observations below a certain point in a normal distribution.
• In applications, it helps forecast probabilities and assess risks.

Recognizing cumulative probabilities allows us to make data-driven conclusions about probabilities related to normally distributed phenomena.

Standard Deviation

Standard deviation is a key statistical measure that quantifies the amount of variation or dispersion in a set of values. It is a vital tool in understanding how spread out the numbers are in a dataset.
A low standard deviation indicates that the data points tend to be close to the mean, whereas a high standard deviation indicates a wide spread of values around the mean.
In our exercise, the standard deviation is given as 8. This tells us that most scores are within 8 points of the mean score of 75.

• Standard deviation provides insights into the reliability and consistency of datasets.
• It's crucial for calculating Z-scores and understanding distribution patterns.

Mastering standard deviation aids in interpreting data symmetry and variance, allowing more accurate decisions based on statistical analysis.

Mean

The mean is the average value of a dataset, and it is one of the most commonly used measures of central tendency. To find the mean, sum up all the values and divide by the total number of values.
In the given exercise, the mean score is 75. This means that if all test scores were added together and averaged, each would be equal to 75.
The mean serves as a key reference or benchmark when evaluating individual data points within a dataset.

• It steers much of the analysis in statistical evaluations, serving as a central hub for understanding data distribution.
• The mean is essential when calculating Z-scores as it forms the baseline from which deviations are measured.
• Knowing the mean allows comparisons across different datasets and interprets individual versus collective performance.

Understanding the mean aids in establishing a foundational understanding of the dataset, guiding further statistical exploration and findings.