Suppose a certain entrance exam has scores that are normally distributed with a mean score of 78 and a standard deviation of 7. What is the probability that a randomly selected student scores higher than 75 on the exam?
well, 75 is 3/7 σ below the mean
so run to your Z table and see what it says.
To find the probability that a randomly selected student scores higher than 75 on the exam, we need to calculate the area under the normal distribution curve to the right of 75.
Step 1: Standardization
First, we need to standardize the score of 75 by using the formula for standardization:
Z = (X - μ) / σ
Where:
Z is the standard score (z-score)
X is the raw score
μ is the mean of the distribution
σ is the standard deviation of the distribution
In this case, X = 75, μ = 78, and σ = 7.
Plugging the values into the formula, we have:
Z = (75 - 78) / 7
Z = -0.43
Step 2: Finding the probability
Next, we need to find the probability associated with the z-score of -0.43 by referring to the z-table or using statistical software.
Using a standard normal distribution table or a statistical calculator, we can find that the area to the left of -0.43 is approximately 0.3325.
Step 3: Calculating the desired probability
To find the probability that a randomly selected student scores higher than 75, we need to subtract the probability of scoring less than or equal to 75 from 1 (since the total area under the normal curve is 1).
P(X > 75) = 1 - P(X ≤ 75)
P(X > 75) = 1 - 0.3325
P(X > 75) ≈ 0.6675
Therefore, the probability that a randomly selected student scores higher than 75 on the exam is approximately 0.6675 or 66.75%.