The mean is I=15.2 and the standard deviation is o=.9. Find the probability that x is between 14.3 and 16.1.
To find the probability that x is between 14.3 and 16.1, we need to calculate the z-scores associated with these values and then use the standard normal distribution table (also known as the z-table) to find the area under the curve between these z-scores.
Step 1: Calculate the z-scores:
The z-score formula is given by z = (x - μ) / σ, where x is the value, μ is the mean, and σ is the standard deviation.
For x = 14.3:
z1 = (14.3 - 15.2) / 0.9 = -1
For x = 16.1:
z2 = (16.1 - 15.2) / 0.9 ≈ 1
Step 2: Use the z-table:
The z-table provides the area under the standard normal curve for different z-scores. Since the z-table is typically given in terms of positive z-scores, we need to find the area between -1 and 1 and then subtract the area to the left of -1 from the area to the left of 1 to get the desired probability.
Using the z-table, the area to the left of z = -1 is 0.1587, and the area to the left of z = 1 is 0.8413.
Step 3: Calculate the desired probability:
To find the probability that x is between 14.3 and 16.1, we subtract the area to the left of -1 from the area to the left of 1:
P(14.3 ≤ x ≤ 16.1) = 0.8413 - 0.1587 = 0.6826
Therefore, the probability that x is between 14.3 and 16.1 is approximately 0.6826, or 68.26%.