A manufacturer produces bearings, but because of variability in the production process, not all of the bearings have the same diameter. The diameters have a normal distribution with a mean of 1.2 centimeters (cm) and a standard deviation of 0.03 cm. The manufacturer has determined that diameters in the range of 1.17 to 1.23 cm are acceptable. What proportion of all bearings falls in the acceptable range? (Round your answer to four decimal places.)

Accepted Solution

Answer:68%Step-by-step explanation:It is given that the diameters of bearing have a normal distribution. Mean = u = 1.2 cmStandard deviation = [tex]\sigma[/tex] = 0.03 cmWe have to find the proportion of values which falls in between 1.17 to 1.23In order to find this we have to convert these values to z-scores first. The formula to calculate z score is:[tex]z=\frac{x- \mu}{\sigma}[/tex]For 1.17:[tex]z=\frac{1.17-1.2}{0.03}=-1[/tex]For 1.23:[tex]z=\frac{1.23-1.2}{0.03}=1[/tex]So, we have to tell what proportion of values fall in between z score of -1 and 1. Since the data have normal distribution we can use empirical rule to answer this question. According to the empirical rule:68% of the values fall within 1 standard deviation of the mean i.e. 68% of the values fall between the z score of -1 and 1. Therefore, the answer to this question is 68%