The random variable X is normally distributed with mean 45 and standard deviation a.Given that P(X>51)=0.288, find the value of a.
http://davidmlane.com/hyperstat/z_table.html
x=(z-u)/sigma
then use the z table.
But remember the z table reads "less than" so you need to do 1 - that value found on the z table : )
To find the value of standard deviation a, we need to use the properties of the standard normal distribution.
Step 1: Convert X to standard normal distribution.
Since X is normally distributed, we can standardize it by converting it to a standard normal distribution using the formula:
Z = (X - μ) / σ
Where Z is the standard normal variable, X is the normally distributed variable, μ is the mean, and σ is the standard deviation.
In this case, X = 51 and μ = 45. We don't know the value of σ yet. So we have:
Z = (51 - 45) / σ
Step 2: Use the cumulative distribution function (CDF) of the standard normal distribution.
To find P(X > 51), we need to calculate the probability that Z is greater than the standardized value of 51. In other words, we need to find P(Z > (51 - 45) / σ).
Given that P(X > 51) = 0.288, the probability that Z is greater than the standardized value of 51 is also 0.288.
Step 3: Use a standard normal distribution table or calculator.
To find the value of Z that corresponds to a cumulative probability of 0.288, we can use a standard normal distribution table or calculator. From the table or calculator, we find that the closest value of Z is approximately 0.89.
Step 4: Solve for σ.
Now we can solve for σ using the formula we derived in Step 2:
0.288 = P(Z > (51 - 45) / σ) = P(Z > 0.89)
However, since we're looking for the probability that Z is greater than 0.89, we need to subtract the cumulative probability corresponding to Z = 0.89 from 1:
0.288 = 1 - P(Z ≤ 0.89)
From the standard normal distribution table or calculator, the cumulative probability corresponding to Z = 0.89 is approximately 0.812. Therefore:
0.288 = 1 - 0.812
0.288 = 0.188
Step 5: Solve for σ.
Now we can solve for σ by substituting the known values into the equation derived in Step 4:
0.188 = (51 - 45) / σ
0.188σ = 6
Divide both sides of the equation by 0.188:
σ = 6 / 0.188
Using a calculator, we find that σ is approximately 31.91.
Therefore, the value of standard deviation a is approximately 31.91.