Assume X~N(1.2, 1.3). Find
P(0.26<|X|<1.68)?
To find the probability P(0.26 < |X| < 1.68), where X follows a normal distribution with mean 1.2 and standard deviation 1.3, we can use the properties of the standard normal distribution.
Step 1: Standardize the values
To use the standard normal distribution table, we need to standardize the given values using the formula:
Z = (X - μ) / σ
where Z is the standardized value, X is the given value, μ is the mean, and σ is the standard deviation.
For the lower bound:
Z1 = (0.26 - 1.2) / 1.3
For the upper bound:
Z2 = (1.68 - 1.2) / 1.3
Step 2: Find the probabilities using the standard normal distribution table
The standard normal distribution table provides the probabilities for Z values. We can find the probabilities for Z1 and Z2 separately.
Let P(Z) represent the probability associated with Z.
P(Z1 < Z < Z2) = P(Z < Z2) - P(Z < Z1)
Step 3: Calculate the probabilities
Using the standard normal distribution table, look up the probabilities for Z1 and Z2.
P(Z < Z2) = Lookup the probability for Z2 in the standard normal distribution table.
P(Z < Z1) = Lookup the probability for Z1 in the standard normal distribution table.
Step 4: Calculate P(0.26 < |X| < 1.68)
Substitute the values of P(Z < Z2) and P(Z < Z1) into the formula:
P(Z1 < Z < Z2) = P(Z < Z2) - P(Z < Z1)
Now, use the values obtained from the standard normal distribution table to calculate the final probability.
Note: If you have access to statistical software like Excel or programming languages like Python, you can directly calculate P(0.26 < |X| < 1.68) using the cumulative distribution function (CDF) of the normal distribution.