The average score on a standardized test is 750 points with a standard deviation of 50 points. If 2,000
students take the test at a local school, how many students do you expect to score between 700 and 800
points?
http://davidmlane.com/hyperstat/z_table.html
To find the number of students expected to score between 700 and 800 points, we need to use the properties of the normal distribution.
Step 1: Calculate the z-scores for 700 and 800.
A z-score measures the number of standard deviations a data point is from the mean.
The formula for calculating the z-score is:
z = (x - μ) / σ
where x is the data point, μ is the mean, and σ is the standard deviation.
For 700 points:
z(700) = (700 - 750) / 50 = -1
For 800 points:
z(800) = (800 - 750) / 50 = 1
Step 2: Use the z-scores to find the corresponding probabilities using a standard normal distribution table or calculator.
The probability corresponds to the area under the curve between the two z-scores.
Using a standard normal distribution table, the probability of getting a score less than -1 is 0.1587.
The probability of getting a score less than 1 is 0.8413.
Step 3: Calculate the expected number of students.
To find the number of students between 700 and 800 points, we subtract the probability of the lower score from the probability of the higher score and multiply by the total number of students.
Expected number of students = (0.8413 - 0.1587) * 2000
Expected number of students = 0.6826 * 2000
Expected number of students = 1365.2
Therefore, we expect approximately 1,365 students to score between 700 and 800 points.
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion using the two Z values and multiply by 2,000.