Question

The diameters of bolts produced by a certain machine are normally distributed with a mean of 0.30 inches and a standard deviation of 0.01 inches. What is the probability of selecting a bolt at random that has a diameter larger than 0.328 inches

Answer 1

Answer:

The probability of selecting a bolt at random that has a diameter larger than 0.328 inches is 0.00256.

Step-by-step explanation:

We are given that the diameters of bolts produced by a certain machine are normally distributed with a mean of 0.30 inches and a standard deviation of 0.01 inches.

Let X = diameters of bolts produced by a certain machine

SO, X ~ N($\mu = 0.30,\sigma^{2} = 0.01^{2}$)

The z-score probability distribution is given by ;

                 Z = $\frac{X-\mu}{\sigma}$ ~ N(0,1)

where, $\mu$ = mean diameter = 0.30 inches

            $\sigma$ = standard deviation = 0.01 inches

The Z-score measures how many standard deviations the measure is away from the mean. After finding the Z-score, we look at the z-score table and find the p-value (area) associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X.

Now, the probability of selecting a bolt at random that has a diameter larger than 0.328 inches is given by = P(X > 0.328 inches)

    P(X > 0.328) = P( $\frac{X-\mu}{\sigma}$ > $\frac{0.328-0.30}{0.01}$ ) = P(Z > 2.80) = 1 - P(Z $\leq$ 2.80)

                                                            = 1 - 0.99744 = 0.00256

The above probability is calculated using z table by looking at value of x = 2.80 in the z table which have an area of 0.99744.

Therefore, probability of selecting a bolt at random that has a diameter larger than 0.328 inches is 0.00256.