Inactive Tutor answered • 04/18/23
To find the probability P(3 ≤ x ≤ 6) for a normally distributed variable x with mean 𝜇 = 4.7 and standard deviation 𝜎 = 1.6, we will use the standard normal distribution (Z-distribution) and the Z-score formula. The Z-score formula is:
Z = (x - 𝜇) / 𝜎
First, we will calculate the Z-scores for x = 3 and x = 6 using the given mean and standard deviation:
Z3 = (3 - 4.7) / 1.6 = -1.0625
Z6 = (6 - 4.7) / 1.6 = 0.8125
Now, we will find the probabilities corresponding to these Z-scores using a standard normal (Z) table or a calculator with a built-in cumulative distribution function (CDF) for the standard normal distribution:
P(Z ≤ -1.0625) = 0.1441
P(Z ≤ 0.8125) = 0.7910
To find the probability P(3 ≤ x ≤ 6), we will subtract the smaller probability from the larger one:
P(3 ≤ x ≤ 6) = P(Z ≤ 0.8125) - P(Z ≤ -1.0625) = 0.7910 - 0.1441 = 0.6469
Thus, the probability P(3 ≤ x ≤ 6) for the given normal distribution is approximately 0.6469, or 64.69%.
Inactive Tutor
I understand that you're looking for an accurate approximation based on the Z-table without using a calculator, considering that we need a 4-decimal place result. Let's see how we can obtain this approximation using linear interpolation: Given the Z-table values, we have P(Z ≤ -1.06) = 0.1446, and P(Z ≤ -1.07) = 0.1423. Since -1.07 < -1.0625 < -1.06, we can use linear interpolation for our approximation. The linear interpolation equation is: Pz - Pz0 = m (z - z0), where m = (Pz1 - Pz0) / (z1 - z0) and Pz = P(Z ≤ z). In this case, z1 = -1.06, z0 = -1.07, Pz1 = 0.1446, and Pz0 = 0.1423. Thus, m = (0.1446 - 0.1423) / (-1.06 + 1.07) = 0.23. Now we can calculate P(Z ≤ z) = 0.1423 + 0.23(z + 1.07), which is a linear approximation for any -1.07 < z < -1.06. Therefore, P(Z ≤ -1.0625) = 0.1423 + 0.23(-1.0625 + 1.07) = 0.144025. As a rule of thumb for linear approximation: considering that the difference between -1.0625 and -1.06 is one-third of the difference between -1.07 and -1.06, we can assume the same ratio for their corresponding probabilities. Thus, divide the difference |P(z ≤ -1.06) - P(z ≤ -1.07)| by four (one distance unit from P(z ≤ -1.06) and three distance units from P(z ≤ -1.07)) and subtract this value from P(z ≤ -1.06) to calculate P(Z ≤ -1.0625).04/18/23
Nikki G.
How did you find 0.1441 on the standard z table? When I looked up -1.0625 on the table i found it corresponded with .14457 and that is where my error was when I worked the problem out originally. Thanks!04/18/23