Question

Given a normal population with \(\mu=25\) and \(\sigma=5\), find the probability that an assumed value of the variable will fall in the interval 20 to 30 .

Short answer

The probability that an assumed value of the variable will fall in the interval 20 to 30 in this normal population is 0.6826 or 68.26%.

Step-by-step solution

First, we'll calculate the z-scores for the interval (20 to 30) using the formula: \[z = \frac{x-\mu}{\sigma}\] Where z is the z-score, x is the assumed value (20 and 30), \(\mu\) is the mean of the population, and \(\sigma\) is the standard deviation of the population. For x = 20: \[z_1 = \frac{20-25}{5} = -1\] For x = 30: \[z_2 = \frac{30-25}{5} = 1\] Now we have the z-scores: \(z_1 = -1\) and \(z_2 = 1\). #Step 2: Calculate the probability using the standard normal table#

We have our z-scores and will use a standard normal table (z-table) to find the probability. Remember that the z-table gives the probability between z = 0 and z = z-score (area under the curve), starting from the mean (in our case, z = 0 corresponds to x = 25). Looking up in the z-table: For \(z_1 = -1\), the area to the left is 0.1587, and For \(z_2 = 1\), the area to the left is 0.8413. #Step 3: Calculate the probability between the z-scores#

Now, to find the probability that the assumed value of the variable will fall between 20 (z-score = -1) and 30 (z-score = 1), we will subtract the area to the left of \(z_1\) from the area to the left of \(z_2\). \[P(z_1 < Z < z_2) = P(Z < z_2) - P(Z < z_1)\] \[P(-1 < Z < 1) = 0.8413 - 0.1587\] #Step 4: Calculate the final probability#

Calculate the probability: \[P(-1 < Z < 1) = 0.8413 - 0.1587 = 0.6826\] The probability that the assumed value of the variable will fall between 20 and 30 is 0.6826 or 68.26%.