If bolt thread length is normally distributed, what is the probability that the thread length of a randomly selected bolt is Within 0.8 SDs of its mean value? Farther than 2.3 SDs from its mean value? Between 1 and 2 SDs from its mean value?

Answer:a) 0.5762b) 0.0214c) 0.2718Step-by-step explanation:It is given that lengths of the bolt thread are normally distributed. So in order to find the required probability we can use the concept of z distribution and z scores.Part a) Probability that length is within 0.8 SDs of the meanWe have to calculate the probability that the length of a bolt thread is within 0.8 standard deviations of the mean. Recall that a z- score tells us that how many standard deviations away a value is from the mean. So, indirectly we are given the z-scores here.Within 0.8 SDs of the mean, means from a score of -0.8  to +0.8. i.e. we have to calculate:P(-0.8 < z < 0.8)We can find these values from the z table.P(-0.8 < z < 0.8) = P(z < 0.8) - P(z < -0.8)= 0.7881 - 0.2119= 0.5762Thus, the probability that the thread length of a randomly selected bolt is within 0.8 SDs of its mean value is 0.5762Part b) Probability that length is farther than 2.3 SDs from the meanAs mentioned in previous part, 2.3 SDs means a z-score of 2.3. 2.3 Standard Deviations farther from the mean, means the probability that z scores is lesser than - 2.3 or greater than 2.3i.e. we have to calculate:P(z < -2.3 or z > 2.3) According to the symmetry rules of z-distribution:P(z < -2.3 or z > 2.3) = 1 - P(-2.3 < z < 2.3)We can calculate P(-2.3 < z < 2.3) from the z-table, which comes out to be 0.9786. So,P(z < -2.3 or z > 2.3) = 1 - 0.9786= 0.0214Thus, the probability that a bolt length is 2.3 SDs farther from the mean is 0.0214Part c) Probability that length is between 1 and 2 SDs from the mean valueBetween 1 and 2 SDs from the mean value can occur both above the mean and below the mean.For above the mean: between 1 and 2 SDs means between the z scores 1 and 2For below the mean: between 1 and 2 SDs means between the z scores -2 and -1i.e. we have to find:P( 1 < z < 2) + P(-2 < z < -1)According to the symmetry rules of z distribution:P( 1 < z < 2) + P(-2 < z < -1) = 2P(1 < z < 2)We can calculate P(1 < z < 2) from the z tables, which comes out to be: 0.1359So,P( 1 < z < 2) + P(-2 < z < -1) = 2 x 0.1359= 0.2718Thus, the probability that the bolt length is between 1 and 2 SDs from its mean value is 0.2718