To determine the percentage of students that take longer than 60 minutes to complete a math competency exam, we need to analyze the given information using normal distribution principles. Let's follow a step by step approach:
### Given Information
- Mean of the distribution ([tex]\( \mu \)[/tex]) = 57.6 minutes
- Standard deviation ([tex]\( \sigma \)[/tex]) = 8 minutes
- Time to evaluate ([tex]\( X \)[/tex]) = 60 minutes
### Step 1: Calculate the Z-Score
First, we need to calculate the Z-score for the time of 60 minutes. The Z-score tells us how many standard deviations a value is from the mean. The formula for the Z-score is:
[tex]\[ Z = \frac{X - \mu}{\sigma} \][/tex]
Substituting the given values into the formula:
[tex]\[ Z = \frac{60 - 57.6}{8} = \frac{2.4}{8} = 0.3 \][/tex]
So, the Z-score for 60 minutes is 0.3.
### Step 2: Find the Cumulative Probability for the Z-Score
Next, we use the Z-score to find the cumulative probability that a student will complete the test within 60 minutes. This can be found using the standard normal distribution table, or a statistical software.
For a Z-score of 0.3, the cumulative probability (or the area under the curve to the left of Z=0.3) is approximately 0.618. This means:
[tex]\[ P(X \leq 60) = 0.618 \][/tex]
### Step 3: Calculate the Complement Probability
To find the percentage of students who take longer than 60 minutes, we need to find the complement of the probability we just found. The complement probability gives us the area under the curve to the right of [tex]\( X = 60 \)[/tex].
[tex]\[ P(X > 60) = 1 - P(X \leq 60) \][/tex]
Substituting the cumulative probability:
[tex]\[ P(X > 60) = 1 - 0.618 = 0.382 \][/tex]
### Step 4: Convert the Probability to Percentage
Finally, we convert the probability into a percentage by multiplying by 100.
[tex]\[ \text{Percentage} = 0.382 \times 100 = 38.2\% \][/tex]
### Answer
Therefore, approximately 38.2% of students take longer than 60 minutes to complete the math competency exam.
### Sketch of the Normal Curve
Here's a sketch to illustrate the problem visually:
1. Draw a standard normal distribution curve.
2. Mark the mean (57.6 minutes) at the center.
3. Locate the point for 60 minutes on the x-axis.
4. Shade the region to the right of 60 minutes to represent the percentage of students taking longer than 60 minutes.
This shaded region represents the 38.2% of students who need more time than 60 minutes to finish their exam.
