Question

In Mrs. Espinol’s class of 25 students, the average score on the final exam is 85, with a standard deviation of seven. How many students scored above \(92 ?\) A. 2 B. 4 C. 14 D. 16

Short answer

4 students scored above 92.

Step-by-step solution

- Understanding the problem

We need to find the number of students who scored above 92 in a class of 25 students. We are given the average score ( = 85) and the standard deviation ( = 7) of the class.

- Calculate Z-score

First, calculate the Z-score of the score 92. The Z-score formula is:\[ Z = \frac{X - \mu}{\sigma} \]where X is the score (92), is the mean (85), and \( \sigma \) is the standard deviation (7).

- Solve for Z-score

Using the formula:\[ Z = \frac{92 - 85}{7} = \frac{7}{7} = 1 \]So, the Z-score for 92 is 1.

- Find corresponding probability

Using the Z-table, find the probability corresponding to the Z-score of 1. The Z-table shows that the probability of Z being less than 1 is approximately 0.8413.

- Calculate the percentage of students above 92

Since the probability of scoring less than 92 is 0.8413, the percentage of students scoring above 92 is:\[ 1 - 0.8413 = 0.1587 \]Converting this to percentage: 0.1587 * 100 ≈ 15.87%.

- Calculate the number of students

To find the number of students who scored above 92, multiply the total number of students by the percentage of students who scored above 92:\[ 0.1587 * 25 ≈ 3.96 \]Since the number of students must be a whole number, we round approximately to 4.

- Select the correct answer

Therefore, the number of students who scored above 92 is approximately 4. The correct answer is B. 4.

Key concepts

mean and standard deviation

In statistics, the mean and standard deviation are two fundamental concepts.
The mean, often referred to as the average, is calculated by adding up all the values in a dataset and then dividing by the number of values.
This provides a central value for the data.
The standard deviation is a measure of how spread out the numbers in a dataset are.
A small standard deviation means that the values are close to the mean, while a large standard deviation indicates that the values are more spread out.
In our exercise, the mean score of Mrs. Espinol’s class is 85, and the standard deviation is 7.
This tells us that while the average score is 85, individual scores can vary around this average by approximately 7 points.

Z-score calculation

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

• X is the value you are evaluating (in this case, a score of 92).
• \(\mu\) is the mean of the dataset (85).
• \(\sigma\) is the standard deviation (7).

Using the provided formula, the Z-score for a score of 92 is calculated as follows:
\[ Z = \frac{92 - 85}{7} = \frac{7}{7} = 1 \]
This Z-score of 1 tells us that a score of 92 is one standard deviation above the mean.

probability and statistics

Probability and statistics are intertwined fields in mathematics that help us understand and interpret data.
Probability measures the likelihood of an event occurring, while statistics involves collecting, analyzing, and interpreting data.
In this context, we use the Z-score to find the probability of a student scoring above 92.
The Z-table gives the probability corresponding to a Z-score of 1, which is 0.8413.
This means there is an 84.13% chance of a score being less than 92.
To find how many students scored above 92, we calculate:
\[ 1 - 0.8413 = 0.1587 \]
Thus, the probability of scoring above 92 is 15.87%.
Multiplying this by the total number of students (25) gives us:
\[ 0.1587 \times 25 \approx 4 \]
This means approximately 4 students scored higher than 92.