a normal distribution has a mean of 22 and a standard deviation of 3. Find the probablility that a randomly selected x-value is in the given interval. Between 19 and 25.
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions related to the two Z scores.
1/6
To find the probability that a randomly selected x-value is in the given interval, we need to calculate the area under the normal distribution curve between the values 19 and 25.
Here's how you can do it step-by-step:
1. Standardize the interval: To use standard tables or calculators, convert the original values 19 and 25 to z-scores. The formula to calculate the z-score is: z = (x - mean) / standard deviation.
For the lower bound (19):
z1 = (19 - 22) / 3 = -1
For the upper bound (25):
z2 = (25 - 22) / 3 = 1
2. Look up the z-scores in the standard normal distribution table (also known as the Z-table) or use a calculator. The table provides the cumulative probability up to a given z-score.
The cumulative probability for z = -1 (lower bound) is P(Z ≤ -1) = 0.1587.
The cumulative probability for z = 1 (upper bound) is P(Z ≤ 1) = 0.8413.
3. Calculate the probability of the interval between 19 and 25:
P(19 ≤ X ≤ 25) = P(Z ≤ 1) - P(Z ≤ -1)
P(19 ≤ X ≤ 25) = 0.8413 - 0.1587
P(19 ≤ X ≤ 25) = 0.6826
So, the probability that a randomly selected x-value is in the interval between 19 and 25 is approximately 0.6826, or 68.26%.