The distribution of SAT scores in math for an incoming class of business students has a mean of 610 and standard deviation of 20. Assume that the scores are normally distributed.

a. Find the probability that an individual’s SAT score is less than 600.
b. Find the probability that an individual’s SAT score is between 590 and 620.
c. Find the probability that an individual’s SAT score is greater than 650.
d. What scores will the top 5% of students have?
e. Find the standardized values for students scoring 540, 600, 650, and 700 on the test. Explain what these mean.

a. The probability that an individual's SAT score is less than 600 is about 0.3085. b. The probability that an individual's SAT score is between 590 and 620 can be found by subtracting the probabilities associated with the z-scores for these values. c. The probability that an individual's SAT score is greater than 650 can be found using the z-score for 650. d. The top 5% of students will have scores corresponding to a certain z-score. e. The standardized values for scores on the test can be calculated using the z-score formula.

Explanation:

a. To find the probability that an individual’s SAT score is less than 600, we need to calculate the z-score. The z-score formula is: z = (x - μ) / σ, where x is the value we're interested in, μ is the mean, and σ is the standard deviation. Plugging in the values, we get: z = (600 - 610) / 20 = -0.5. Using a z-table or a calculator, we can find that the probability is about 0.3085.

b. To find the probability that an individual’s SAT score is between 590 and 620, we need to find the z-scores for both values. The z-score for 590 is: z = (590 - 610) / 20 = -1. The z-score for 620 is: z = (620 - 610) / 20 = 0.5. Using the z-table or calculator, we can find the probabilities associated with these z-scores and subtract them to find the desired probability.

c. To find the probability that an individual’s SAT score is greater than 650, we need to find the z-score for 650. The z-score is: z = (650 - 610) / 20 = 2. Using the z-table or calculator, we can find the probability associated with this z-score.

d. To find the scores that will correspond to the top 5% of students, we need to find the z-score that corresponds to this probability. This z-score can be found using the inverse of the z-table or a calculator. Once we have the z-score, we can find the corresponding SAT score using the z-score formula.

e. To find the standardized values for students scoring 540, 600, 650, and 700 on the test, we can use the z-score formula. Plugging in the values, we can calculate the z-scores for each score.

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