Problem 42 - Algebra 2

Question

A set of 400 test scores is normally distributed with a mean of 75 and a standard deviation of 8 . What percent of the test scores lie between 67 and 83\(?\)

Step-by-Step Solution

Answer

Approximately 68.26% of the test scores lie between 67 and 83.

Step 1: Understand the Problem

We need to calculate the percentage of test scores that fall within the interval of 67 to 83 from a normally distributed set. The distribution has a mean (\( \mu \)) of 75 and a standard deviation (\( \sigma \)) of 8.

Step 2: Calculate Z-Scores

A Z-score represents the number of standard deviations a data point is from the mean. To find the Z-scores, use the formula: \[ Z = \frac{X - \mu}{\sigma} \]where \( X \) is the value for which you're calculating the Z-score.Let's calculate Z-scores for both 67 and 83:For 67: \[ Z = \frac{67 - 75}{8} = \frac{-8}{8} = -1 \]For 83: \[ Z = \frac{83 - 75}{8} = \frac{8}{8} = 1 \]

Step 3: Use Z-Score Table

Using a Z-score table (or a standard normal distribution table), we find the probabilities associated with these Z-scores. For Z = -1, the probability is approximately 0.1587. For Z = 1, the probability is approximately 0.8413.

Step 4: Calculate the Percentage between Z-Scores

To find the percentage of test scores between the Z-scores of -1 and 1, subtract the lower Z-score probability from the higher Z-score probability:\[ P(67 < X < 83) = 0.8413 - 0.1587 = 0.6826 \]Thus, 68.26% of the test scores lie between 67 and 83.

Key Concepts

Understanding Standard Deviation

Standard deviation is a measure that quantifies the amount of variation or dispersion in a set of data values. In the context of a normal distribution, it tells you how much the values deviate from the mean, which is the central point. The formula for standard deviation is given by \( \sigma = \sqrt{\frac{1}{N}\sum_{i=1}^{N}(X_i - \mu)^2} \),where \( \sigma \) is the standard deviation, \( N \) is the total number of observations, \( X_i \) are the individual data points, and \( \mu \) is the mean of the data.

• In our example, a standard deviation of 8 indicates that most of the test scores are within 8 points of the mean.
• If the standard deviation were smaller, the test scores would be clustered more tightly around the mean.
• A larger standard deviation would signify more spread in the scores.

Understanding standard deviation is crucial as it helps in interpreting the reliability of the average (mean) calculated from the dataset. It tells us the confidence level at which we can predict outcomes based on the dataset.

Explaining Z-Score

A Z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. It is expressed in terms of standard deviations from the mean. The formula for calculating a Z-score is:\[ Z = \frac{X - \mu}{\sigma} \]where \( X \) is the raw score, \( \mu \) is the mean of the distribution, and \( \sigma \) is the standard deviation.

• A Z-score of 0 indicates the score is exactly at the mean.
• A positive Z-score means the data point is above the mean.
• A negative Z-score signifies that the data is below the mean.

Z-scores are handy because they allow direct comparison between data points from different distributions. They also help in determining the probability of occurrence of scores within a normal distribution, as seen in our calculation of test scores between 67 and 83 points.

Details on Mean Calculation

The mean is a measure of central tendency, often referred to as the 'average.' It is calculated by adding up all the numbers in your data set and then dividing by the count of the numbers. The formula is:\( \mu = \frac{\sum X_i}{N} \),where \( \sum X_i \) is the sum of all data points, and \( N \) is the number of data points.

• The mean represents where the center of the data lies.
• It is sensitive to extreme values (outliers).
• In a normal distribution, the mean is located at the center, and about half of the data falls below and above this point.

Understanding mean calculation is necessary as it serves as a foundation for other statistical measurements, such as standard deviation and Z-scores. In our exercise, the mean of 75 serves as the central pivot for the distribution of test scores.

Probability Distribution Insights

Probability distribution describes how the values of a random variable are distributed. In our context, we deal with the normal distribution – a symmetrical, bell-shaped curve that is characterized by its mean and standard deviation.

• The total area under the curve of a probability distribution sums up to 1, representing a total probability of 100%.
• A normal distribution is defined by its mean (center) and standard deviation (spread).
• Segmenting the curve based on Z-scores allows us to compute the probability of a data point falling within a specific range.

In our exercise, the normal distribution explains the probability of test scores falling between certain score values. That is, 68.26% of scores are expected to lie between 67 and 83 within the dataset's distribution, emphasizing the normal distribution's predictive power.