Test scores on a university admissions test are normally distributed, with a mean of 500 and a standard deviation of 100.
4 marks a. What is the probability that a randomly selected applicant scores between 425 and 575?
4 marks b. What is the probability that a randomly selected applicant scores 625 or more?
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions/probabilities related to the Z scores.
Test scores on a university admissions test are normally distributed, with a mean of 500 and a standard deviation of 100.
4 marks a. What is the probability that a randomly selected applicant scores between 425 and 575?
4 marks b. What is the probability that a randomly selected applicant scores 625 or more?
To find the probability that a randomly selected applicant scores between 425 and 575, or scores 625 or more, we need to use the concept of the standard normal distribution. The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1.
To solve this problem, we need to convert the given values into z-scores, which represent the number of standard deviations an observation is away from the mean in either direction.
a. To find the probability that a randomly selected applicant scores between 425 and 575, we need to find the area under the curve between these two values. First, we need to convert these values into z-scores.
Z-score of 425: (425 - 500) / 100 = -0.75
Z-score of 575: (575 - 500) / 100 = 0.75
Next, we can use a standard normal distribution table or a statistical calculator to find the probability associated with these z-scores. The probability associated with a z-score of -0.75 is 0.2266, and the probability associated with a z-score of 0.75 is 0.7734.
Finally, we can subtract the probability of the lower z-score from the probability of the higher z-score to find the probability that a randomly selected applicant scores between 425 and 575.
Probability = 0.7734 - 0.2266 = 0.5468
Therefore, the probability that a randomly selected applicant scores between 425 and 575 is 0.5468, or 54.68%.
b. To find the probability that a randomly selected applicant scores 625 or more, we need to find the area under the curve to the right of the z-score corresponding to 625.
Z-score of 625: (625 - 500) / 100 = 1.25
Using a standard normal distribution table or a statistical calculator, we can find the probability associated with a z-score of 1.25, which is 0.8944.
Therefore, the probability that a randomly selected applicant scores 625 or more is 0.8944, or 89.44%.