Z = (score-mean)/SD

Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to the Z scores.

# The SAT scores of a certain population of students are approximatley normally distributed with a mean and standard deviation of 1000 and 200 points, respectively. the probabilalty that a student selected at random from the population has an SAT score between 800 and 1200 is?

PsyDAG
answered

11 years ago

11 years ago

Explain Bot
answered

1 year ago

1 year ago

To find the probability that a student selected at random from the population has an SAT score between 800 and 1200, we need to calculate the area under the normal distribution curve between these two scores.

Step 1: Convert the scores to z-scores.

Z-score = (X - Mean) / Standard Deviation

For 800:

Z1 = (800 - 1000) / 200 = -1

For 1200:

Z2 = (1200 - 1000) / 200 = 1

Step 2: Look up the z-scores in the standard normal distribution table. The table will give us the corresponding probabilities.

The cumulative probability associated with a z-score of -1 is 0.1587, and the cumulative probability associated with a z-score of 1 is 0.8413.

Step 3: Find the probability between these two z-scores.

P(Between 800 and 1200) = P(Z1 < Z < Z2)

P(Between 800 and 1200) = P(Z < 1) - P(Z < -1)

P(Between 800 and 1200) = 0.8413 - 0.1587

P(Between 800 and 1200) = 0.6826

Therefore, the probability that a student selected at random from the population has an SAT score between 800 and 1200 is approximately 0.6826 or 68.26%.

Step 1: Convert the scores to z-scores.

Z-score = (X - Mean) / Standard Deviation

For 800:

Z1 = (800 - 1000) / 200 = -1

For 1200:

Z2 = (1200 - 1000) / 200 = 1

Step 2: Look up the z-scores in the standard normal distribution table. The table will give us the corresponding probabilities.

The cumulative probability associated with a z-score of -1 is 0.1587, and the cumulative probability associated with a z-score of 1 is 0.8413.

Step 3: Find the probability between these two z-scores.

P(Between 800 and 1200) = P(Z1 < Z < Z2)

P(Between 800 and 1200) = P(Z < 1) - P(Z < -1)

P(Between 800 and 1200) = 0.8413 - 0.1587

P(Between 800 and 1200) = 0.6826

Therefore, the probability that a student selected at random from the population has an SAT score between 800 and 1200 is approximately 0.6826 or 68.26%.