The average score on a standardized test is 500 points with a standard deviation of 50 points. What is
the probability that a student scores between 450 and 600 on the standardized test?
http://davidmlane.com/hyperstat/z_table.html
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the two Z scores and calculate the proportion between them.
To find the probability that a student scores between 450 and 600 on the standardized test, we need to calculate the area under the normal curve between those two scores.
Step 1: Calculate the z-scores for the lower and upper limits.
The z-score represents the number of standard deviations a particular value is from the mean.
Z1 = (450 - 500) / 50
Z2 = (600 - 500) / 50
Step 2: Look up the z-scores in the Z-table.
The Z-table provides the area under the normal curve to the left of a given z-score.
Using the Z-table, we find that the area to the left of Z1 is approximately 0.1587, and the area to the left of Z2 is approximately 0.8413.
Step 3: Calculate the probability between the two limits.
To find the probability between the two limits, we subtract the area to the left of the lower limit from the area to the left of the upper limit.
P = area between Z1 and Z2
P = area to the left of Z2 - area to the left of Z1
P = 0.8413 - 0.1587
Step 4: Calculate the final probability.
P = 0.6826
Therefore, the probability that a student scores between 450 and 600 on the standardized test is approximately 0.6826, or 68.26%.
To find the probability that a student scores between 450 and 600, we need to standardize these scores by converting them into z-scores.
The z-score formula is given by:
z = (x - μ) / σ
Where:
- z is the z-score,
- x is the value of the score,
- μ is the mean (average) score,
- σ is the standard deviation.
First, let's calculate the z-score for a score of 450:
z1 = (450 - 500) / 50
= -0.5
Next, let's calculate the z-score for a score of 600:
z2 = (600 - 500) / 50
= 2.0
Now that we have the z-scores, we can use a z-table or a calculator to find the probabilities associated with these z-scores.
The z-table provides the area under the standard normal curve for different z-scores. Since the scores on the standardized test follow a normal distribution, we can use the z-table in this case.
Using the z-table:
- The probability associated with z1 = -0.5 is approximately 0.3085.
- The probability associated with z2 = 2.0 is approximately 0.9772.
To find the probability that a student scores between 450 and 600, we need to find the area under the curve between these two z-scores.
P(450 ≤ x ≤ 600) = P(z1 ≤ z ≤ z2)
To calculate this probability, we subtract the cumulative probability associated with z1 from the cumulative probability associated with z2.
P(450 ≤ x ≤ 600) = P(z ≤ z2) - P(z ≤ z1)
= P(z ≤ 2.0) - P(z ≤ -0.5)
= 0.9772 - 0.3085
= 0.6687
Therefore, the probability that a student will score between 450 and 600 on the standardized test is approximately 0.6687, or 66.87%.