In an all boys school, the heights of the student body are normally distributed with a mean of 70 inches and a standard deviation of 5 inches. What is the probability that a randomly selected student will be between 57 and 63 inches tall, to the nearest thousandth?
To find the probability that a randomly selected student will be between 57 and 63 inches tall, we need to find the z-scores for both heights and then use the standard normal distribution table.
First, we need to find the z-score for 57 inches:
z = (x - mean) / standard deviation
z = (57 - 70) / 5
z = -13 / 5
z = -2.6
Next, we need to find the z-score for 63 inches:
z = (x - mean) / standard deviation
z = (63 - 70) / 5
z = -7 / 5
z = -1.4
Now, we can use the standard normal distribution table to find the probabilities associated with these z-scores.
For z = -2.6, the probability is 0.0047
For z = -1.4, the probability is 0.0808
To find the probability that a randomly selected student will be between 57 and 63 inches tall, we subtract these two probabilities:
0.0808 - 0.0047 = 0.0761
Therefore, the probability that a randomly selected student will be between 57 and 63 inches tall is approximately 0.0761 or 7.61%.