As per the concept of normal distribution, the two diameters of the bolt that has separate the top 3% and the bottom 3% are 5.2836 mm and 5.0764 mm respectively .
Normal distribution:
Normal distribution means the probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean and it has the bell shaped curve.
Given,
The diameters of bolts produced in a machine shop are normally distributed with a mean of 5.18 millimeters and a standard deviation of 0.07 millimeters.
Here we need to find the two diameters that separate the top 3% and the bottom 3% and these diameters could serve as limits used to identify which bolts should be rejected. we have also round off the result into nearest hundredth.
Here we have the given question with the values of,
The value of Mean is 5.18 millimeters
The value of Standard deviation is 0.07 millimeters.
Here we have to consider the following two cases, they are.
Case 1
We need to find x₁ such that that is P(X > x₁) = 3% when we convert it into decimal then we get P (X > x₁) = 0.03
So, as per the probability,
P (X ≤ x₁) = 1 - 0.03 = 0.97
So, z corresponding to the value of p = 0.97 is 1.48
Apply the given values to the formula of normal distribution then we get,
=> 1.48 = (x₁ - 5.18) /0.07
=> 0.1036 = x₁ - 5.18
Therefore, the value of x₁ is 5.2836
So, the diameter of the both with separate top 3% is 5.2836.
Case 2:
Next we have to take P(X < x₂) = 3% when we convert it into decimal then we get P (X < x₂) = 0.03
So, as per the given probability, the value of
P (X ≤ x₂) = 1 - 0.03 = 0.97
So, z corresponding to but here we have to use the negative because of the less than concept p = 0.97 is -1.48
Apply the values on the formula of normal distribution then we get,
-1.48 = (x₂ - 5.18) /0.07
Move the value to the other side
-0.1036 = x₂ - 5.18
When we simplify it,
x₂ = 5.0764
So, the diameter of the both with separate bottom 3% is 5.0764
To know more about Normal distribution here.
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