Scores on an English test are Normally distributed with a mean of 78% and a standard deviation of 4.5%. Find the Value of the first quartile.
To find the value of the first quartile, we need to follow these steps:
Step 1: Convert the percentage values to decimal form. For the mean, 78% becomes 0.78, and for the standard deviation, 4.5% becomes 0.045.
Step 2: Calculate the z-score corresponding to the first quartile. The first quartile is the 25th percentile, which means 25% of the scores are below it. We can find the z-score using the standard Normal distribution table or a calculator.
Using the standard Normal distribution table, the closest value we find to 0.25 is 0.5987, which corresponds to a z-score of -0.674.
Step 3: Calculate the value of the first quartile (Q1) using the z-score formula:
Q1 = Mean + (z * standard deviation)
Q1 = 0.78 + (-0.674 * 0.045)
Q1 ≈ 0.78 - 0.030405 ≈ 0.749595
Therefore, the value of the first quartile is approximately 0.7496, or you can convert it back to a percentage by multiplying it by 100.
Note: In this context, it is important to understand that finding percentiles assumes a continuous distribution, while test scores are typically discrete. However, since the number of scores is usually large, we can still approximate the percentiles using the Normal distribution.