How Many Students Score Between 60 and 80 on a Math Exam with a Mean of 70 and Standard Deviation of 10?

The mean mark on a math exam was 70 with a standard deviation of 10. If 200 students wrote the exam, how many would be expected to score between 60 and 80? This problem can be solved using the properties of the normal distribution, which is a fundamental concept in statistics. Let's break down the steps involved in finding the expected number of students who scored within this range.

Understanding the Normal Distribution

The normal distribution, also known as the Gaussian distribution, is a continuous probability distribution that is symmetric around the mean. In the context of this problem, the mean mark on the exam is 70 (μ 70), and the standard deviation is 10 (σ 10).

Calculating Z-Scores

To use the properties of the normal distribution to find the expected number of students, we need to standardize the scores to z-scores. The z-score represents how many standard deviations an element is from the mean.

Step 1: Calculate the Z-Scores

We calculate the z-score for the lower and upper bounds of the score range:
For 60:
z1 (X - μ) / σ (60 - 70) / 10 -1
For 80:
z2 (X - μ) / σ (80 - 70) / 10 1

Finding the Area Under the Normal Curve

The next step is to find the area under the normal curve between the two z-scores. This can be done using standard normal distribution tables or a calculator.

Step 2: Find the Area Under the Normal Curve

The area to the left of z -1 is approximately 0.1587. The area to the left of z 1 is approximately 0.8413. The area between z -1 and z 1 is the difference between these two areas:

P(z -1 to z 1) P(z

Calculating the Expected Number of Students

Since the distribution is normal and the area between z -1 and z 1 represents the probability that a student scores between 60 and 80, we can calculate the expected number of students:

Expected number of students Total number of students times; Area between z -1 and z 1 200 times; 0.6826 approx; 136.52

Since the number of students must be a whole number, we round this to approximately 137 students.

Conclusion

Based on the properties of the normal distribution, approximately 137 students are expected to score between 60 and 80 on the math exam.

Bonus Tip: If you need to find more probabilities or use other statistical tools, consider using the Breatter App, which provides step-by-step solutions to probability distributions, confidence intervals, and hypothesis testing.

Note: For a continuous distribution, the probability of a specific score is zero. However, the range of scores between 60 and 80 covers a significant portion of the distribution, as reflected by the 68.269% probability.

By understanding the normal distribution and how to calculate z-scores, students and educators can better interpret and analyze test results.