A component is rejected if its has a diameter less than 70mm, reworked if the diameter is between 70 and 73.5mm or more than 78.5mm. The diameter of a component is normally distributed with a mean 76mm and standard deviation 2.5mm. In a batch of 400 components, determine the number of components:
(1) that are rejected,
(2) that are accepted
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to the Z scores. Multiply by 400.
For above question
To determine the number of components that are rejected and accepted, we need to find the proportion of components that fall into each category based on the given criteria.
First, let's calculate the z-scores for the cutoff values. The z-score is calculated by subtracting the mean from the value of interest and then dividing by the standard deviation.
For the rejection criteria:
Lower cutoff value (70mm):
z = (70 - 76) / 2.5 = -2.4
For the rework criteria:
Lower cutoff value (70mm):
z = (70 - 76) / 2.5 = -2.4
Upper cutoff value (73.5mm):
z = (73.5 - 76) / 2.5 = -1.4
Upper cutoff value (78.5mm):
z = (78.5 - 76) / 2.5 = 1.0
The rejection region consists of values lower than -2.4 and higher than 1.0. The rework region consists of values between -2.4 and -1.4.
To find the proportions of components falling in the rejection, rework, and acceptance regions, we can use the cumulative distribution function (CDF) of the normal distribution.
(1) Number of components rejected:
P(Z < -2.4) + P(Z > 1.0)
Using a standard normal distribution table or a calculator, we can find the probabilities associated with these z-scores.
P(Z < -2.4) = 0.0082
P(Z > 1.0) = 0.1587
The total proportion of rejected components is 0.0082 + 0.1587 = 0.1669.
To find the number of rejected components out of 400, multiply the proportion by the total number:
Number of components rejected = 0.1669 * 400 = 66.76 ≈ 67 (rounded to the nearest whole number)
Therefore, there are approximately 67 components rejected.
(2) Number of components accepted:
P(-2.4 < Z < -1.4)
Using a normal distribution table or calculator, we can find this probability.
P(-2.4 < Z < -1.4) = 0.0808
The proportion of components that fall into the acceptance region is 0.0808.
Number of components accepted = 0.0808 * 400 = 32.32 ≈ 32 (rounded to the nearest whole number)
Therefore, there are approximately 32 components accepted.