LSAT test scores are normally distributed with a mean of 510 and a standard deviation of 110. Find the probability that a randomly chosen test-taker will score between 300 and 580. (Round your answer to four decimal places.)
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion between the two Z scores.
To find the probability that a randomly chosen test-taker will score between 300 and 580, we need to calculate the z-scores for both values and then look up the corresponding probabilities from the standard normal distribution table.
Step 1: Calculate the z-score for the lower boundary of 300:
z1 = (300 - mean) / standard deviation
= (300 - 510) / 110
= -2.182
Step 2: Calculate the z-score for the upper boundary of 580:
z2 = (580 - mean) / standard deviation
= (580 - 510) / 110
= 0.636
Step 3: Look up the probabilities corresponding to the z-scores from the standard normal distribution table. The probability for the lower boundary is P(Z < -2.182) and the probability for the upper boundary is P(Z < 0.636).
Using the standard normal distribution table or a calculator, we find:
P(Z < -2.182) = 0.0151
P(Z < 0.636) = 0.7357
Step 4: Calculate the probability of being between the two boundaries:
P(300 < X < 580) = P(Z < 0.636) - P(Z < -2.182)
= 0.7357 - 0.0151
= 0.7206
Therefore, the probability that a randomly chosen test-taker will score between 300 and 580 is 0.7206 (rounded to four decimal places).