Scores on a test are normally distributed and have a mean of 75 and a standard deviation of 4.2. What score does a student need in order to rank in the top 20%?
play around here to learn how the Z table stuff works:
http://davidmlane.com/hyperstat/z_table.html
To find the score a student needs in order to rank in the top 20%, we need to find the z-score corresponding to the top 20% and then convert it back to the original data scale using the formula:
z = (x - μ) / σ
where:
z = z-score
x = score
μ = mean
σ = standard deviation
We can use the standard normal distribution table, or a calculator with statistical functions to find the z-score corresponding to the top 20%.
First, we need to find the z-score corresponding to the top 20%:
Step 1: Convert the top 20% to a decimal: 20% = 0.20
Step 2: Since we are looking for the top 20%, subtract 0.20 from 1 to get 0.80.
Step 3: Look up the z-score that corresponds with 0.80. From the standard normal distribution table, the z-score is approximately 0.84.
Now, we can substitute the values into the formula and solve for x:
0.84 = (x - 75) / 4.2
To isolate x, we can multiply both sides of the equation by 4.2:
0.84 * 4.2 = x - 75
3.528 = x - 75
To solve for x, add 75 to both sides of the equation:
3.528 + 75 = x
x ≈ 78.53
Therefore, a student needs a score of approximately 78.53 to rank in the top 20%.