⦁ Find the probability that a data value picked at random from a normal population will have a standard score (z) that lies between the following pairs of z-values.
⦁ z = 0 to z = 2.10
⦁ X
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability between the Z score of 2.1 and the mean (Z = 0).
To find the probability that a data value picked at random from a normal population will have a standard score (z) that lies between two given z-values, we need to use the standard normal distribution table or a statistical software.
The standard normal distribution table provides the area under the curve to the left of a given z-value. Since the standard normal distribution is symmetrical, we can find the probability for the range between two z-values by subtracting the area to the left of the lower z-value from the area to the left of the higher z-value.
In this case, we need to find the probability that z lies between 0 and 2.10. We find the area to the left of each z-value using the standard normal distribution table:
For z = 0: The area to the left of z = 0 is 0.5000.
For z = 2.10: The area to the left of z = 2.10 is 0.9821.
To find the probability between z = 0 and z = 2.10, we subtract the area to the left of z = 0 from the area to the left of z = 2.10:
P(0 ≤ z ≤ 2.10) = Area to the left of z = 2.10 - Area to the left of z = 0
= 0.9821 - 0.5000
= 0.4821
So, the probability that a data value picked at random from a normal population will have a standard score that lies between z = 0 and z = 2.10 is approximately 0.4821 or 48.21%.