Final answer:
The first quartile for the scores on the exam is approximately 73.93. Another name for the first quartile is Q1.
Step-by-step explanation:
The first quartile is a measure of central tendency that represents the value below which 25% of the data falls. Another name for the first quartile is Q1. To find the first quartile for scores on an exam that is normally distributed with a mean of 80 and variance of 81, we need to calculate the z-score for Q1 and then use the z-score to find the corresponding value using the standard normal distribution table.
To calculate the z-score for Q1, we use the formula z = (X - μ) / σ, where X is the value we want to find the z-score for, μ is the mean, and σ is the standard deviation. In this case, X = Q1, which is what we want to find, μ = 80, and σ = √81 = 9.
Using the z-score formula, we have z = (Q1 - 80) / 9. We can rearrange this formula to solve for Q1: Q1 = 80 + 9z.
Next, we need to find the z-score that corresponds to the first quartile. For that, we need to find the area to the left of the first quartile in the standard normal distribution. Since the first quartile represents 25% of the data, the area to the left of Q1 is also 25%. Looking up this area in the standard normal distribution table, we find that the z-score corresponding to the first quartile is approximately -0.674.
Finally, plugging this z-score into the equation Q1 = 80 + 9z, we have Q1 = 80 + 9(-0.674) = 80 - 6.066 = 73.934. Therefore, the first quartile for the scores on the exam is approximately 73.93.