Scores on a certain nationwide college entrance examination follow a normal distribution with a mean of 400 and a standard deviation of 100. Find the probability that a student will score over 500.
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion related to the Z score.
To find the probability that a student will score over 500, we need to calculate the area under the normal distribution curve to the right of 500. Here's how you can find the answer step-by-step:
Step 1: Standardize the score
We need to convert the score of 500 into a standardized Z-score. The formula for standardizing a score is:
Z = (X - μ) / σ
Where:
Z = Standardized score (Z-score)
X = Score value
μ = Mean of the distribution
σ = Standard deviation of the distribution
In this case, X = 500, μ = 400, and σ = 100. Plugging these values into the formula, we get:
Z = (500 - 400) / 100 = 1
Step 2: Find the cumulative probability
Now that we have the Z-score, we can use a standard normal distribution table or a statistical calculator to find the cumulative probability associated with a Z-score of 1. If you are using a table, look for the area to the right of 1. Find the corresponding value, or use a calculator to find this probability. Let's denote this probability as P(Z > 1).
Step 3: Calculate the final probability
Since we want to find the probability that a student will score over 500, we subtract the cumulative probability obtained in step 2 from 1. This gives us:
P(X > 500) = 1 - P(Z < 1)
Remember that P(Z > 1) = 1 - P(Z < 1) because the total area under the distribution curve is 1. Therefore, we can simplify our expression as:
P(X > 500) = 1 - P(Z < 1)
Now, you can either use a standard normal distribution table or a calculator to find P(Z < 1) and subtract it from 1 to get the final probability.
Note: For convenience, you can also use online calculators that provide the probability directly by inputting the mean, standard deviation, and the desired value.