The distriubution of the SAT scores in math for an incoming class of business students has a mean of 580 and standard deviation of 25. Assume that the scores are normally distributed.

a. Find the probability that an individual's SAT score is less that 550.
b. Find the probability that an individuals's SAT score is between 560 and 600.
c. Find the probability that an individuals's SAT score is greater than 620.
d. What percentages of students will have scored better than 700?
e. Find the standardized values for students scoring 500, 600, and 700 on the test

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a. To find the probability that an individual's SAT score is less than 550, we need to find the area under the normal curve to the left of 550.

First, let's calculate the z-score using the formula:

z = (X - μ) / σ

where X is the value (550 in this case), μ is the mean (580), and σ is the standard deviation (25).

z = (550 - 580) / 25 = -1.2

Next, we need to find the cumulative probability associated with the z-score of -1.2. We can look up this probability in a standard normal distribution table or use a calculator to get the value.

Using a calculator, we find that the cumulative probability is approximately 0.1151. Therefore, the probability that an individual's SAT score is less than 550 is approximately 0.1151.

b. To find the probability that an individual's SAT score is between 560 and 600, we need to find the area under the normal curve between these two values.

First, let's calculate the z-scores for each value:
z1 = (560 - 580) / 25 = -0.8
z2 = (600 - 580) / 25 = 0.8

The probability of being within a certain range is the difference between the cumulative probabilities at the upper and lower z-scores. We can calculate this using a standard normal distribution table or a calculator.

Using a calculator, we find that the cumulative probability at z1 is approximately 0.2119, and the cumulative probability at z2 is approximately 0.7881. Therefore, the probability that an individual's SAT score is between 560 and 600 is approximately 0.7881 - 0.2119 = 0.5762.

c. To find the probability that an individual's SAT score is greater than 620, we need to find the area under the normal curve to the right of 620.

First, let's calculate the z-score:
z = (620 - 580) / 25 = 1.6

Using a calculator, we find that the cumulative probability at z is approximately 0.9452. Therefore, the probability that an individual's SAT score is greater than 620 is approximately 1 - 0.9452 = 0.0548.

d. To find the percentage of students who scored better than 700, we need to calculate the cumulative probability at the z-score corresponding to 700.

First, let's calculate the z-score:
z = (700 - 580) / 25 = 4.8

Using a calculator, we find that the cumulative probability at z is approximately 1. Therefore, 100% of students will have scored better than 700.

e. To find the standardized values for students scoring 500, 600, and 700 on the SAT, we need to calculate the z-scores for each value using the formula:

z = (X - μ) / σ

where X is the value, μ is the mean, and σ is the standard deviation.

For 500:
z = (500 - 580) / 25 = -3.2

For 600:
z = (600 - 580) / 25 = 0.8

For 700:
z = (700 - 580) / 25 = 4.8

To answer these questions, we will use the normal distribution and the properties of z-scores. The formula to standardize a value into a z-score is given by:

z = (x - μ) / σ

where x is the value, μ is the mean, and σ is the standard deviation.

a. To find the probability that an individual's SAT score is less than 550, we need to find the area under the normal curve to the left of 550. Since the scores are normally distributed, we can use the z-score formula to standardize the value of 550:

z = (550 - 580) / 25
z = -30 / 25
z = -1.2

Using a standard normal distribution table or a calculator, we can find the corresponding cumulative probability for a z-score of -1.2. The probability is approximately 0.1151, or 11.51%.

b. To find the probability that an individual's SAT score is between 560 and 600, we need to find the area under the normal curve between these two values. We can standardize both values and then find the difference between their cumulative probabilities:

z1 = (560 - 580) / 25
z1 = -20 / 25
z1 = -0.8

z2 = (600 - 580) / 25
z2 = 20 / 25
z2 = 0.8

Using a standard normal distribution table or a calculator, we can find the cumulative probabilities for these two z-scores. The corresponding probabilities are approximately 0.2119 and 0.7881, respectively. The probability of an SAT score between 560 and 600 is the difference between these two probabilities:

0.7881 - 0.2119 = 0.5762, or 57.62%.

c. To find the probability that an individual's SAT score is greater than 620, we need to find the area under the normal curve to the right of 620. We can standardize the value and then find the cumulative probability:

z = (620 - 580) / 25
z = 40 / 25
z = 1.6

Using a standard normal distribution table or a calculator, we can find the cumulative probability for a z-score of 1.6. The probability is approximately 0.9452, or 94.52%.

d. To find the percentage of students who scored better than 700, we need to find the area under the normal curve to the right of 700. First, we need to calculate the z-score for 700:

z = (700 - 580) / 25
z = 120 / 25
z = 4.8

Using a standard normal distribution table or a calculator, we can find the cumulative probability for a z-score of 4.8. However, the standard normal distribution table typically only goes up to 3.49. For z-scores beyond that, the probability is essentially 1, or 100%. Therefore, the percentage of students who scored better than 700 is approximately 100%.

e. To find the standardized values for students scoring 500, 600, and 700 on the test, we will use the z-score formula.

For a score of 500:
z = (500 - 580) / 25
z = -80 / 25
z = -3.2

For a score of 600:
z = (600 - 580) / 25
z = 20 / 25
z = 0.8

For a score of 700:
z = (700 - 580) / 25
z = 120 / 25
z = 4.8

Therefore, the standardized values for scores of 500, 600, and 700 are -3.2, 0.8, and 4.8, respectively.