The diameters of bolts produced by a certain machine are normally distributed with a population mean of 11.0 milimeters (mm) and a population standard deviation of 0.2 mm. What is the probability that a randomly selected bolt will have a diameter within the lower and upper tolerances of 10.44 mm and 11.56 mm? Use 4 decimal places in answering.
Accepted Solution
A:
Answer: The probability that a randomly selected bolt will have a diameter within the lower and upper tolerances of 10.44 mm and 11.56 mm is0.9949 (4 d.p)Step-by-step explanation:FIRST STEP: Determining the mean, standard deviation and the probability we are looking for.The mean, μ= 11.0mm, The standard deviation, σ= 0.2mm,Pr (10.44mm < x < 11.56mm) [This is the probability we are looking for between the lower and upper tolerances]SECOND STEP: Drawing the normal distribution curve and indicating the information we are looking for, also finding the z-scores from the table. [Kindly note you will find attached diagrams below to aid with this]Z₁ = (x₁ - μ)/σ = (10.44 - 11)/0.2 = -2.8Z₂ = (x₂ - μ)/σ = (11.56 - 11)/0.2 = 2.8These are the z-scores we are looking for. (Kindly remember that the z-scores apply to the areas shaded to the left)THIRD STEP: Finding the probabilities from the standard normal distribution curve and finding the probability of x. (Attached photos would also be available)Pr(Z₁) = 0.00256Pr(Z₂) = 0.99744 (These are all for areas to the left)So the Pr (10.44mm < x < 11.56mm) = Pr (Z₁ < x < Z₂) Pr(x) = 0.99744 - 0.00256 (subtraction is necessary because of the fact that the area of Z₂ covers Z₁ and we are looking for only the area between the two z-scores) = 0.99488 = 0.9949(4 d.p)