The lengths of nails produced in a factory are normally distributed with a mean of 4.98 centimeters and a standard deviation of 0.05 centimeters. Find the two lengths that separate the top 5% and the bottom 5%. These lengths could serve as limits used to identify which nails should be rejected. Round your answer to the nearest hundredth, if necessary. USING EXCEL

Given that the lengths of nails produced in a factory are normally distributed with a mean of 4.98 centimeters and a standard deviation of 0.05 centimeters.

If X is the length of nail then X is N(4.98,0.05)

Or [tex]\frac{x-4.98}{0.05}[/tex] is N(0,1)

For std normal variate top and bottom 5% are having z values as

±1.645

i.e. bottom with negative sign and top 5% with positive sign

Corresponding X values we have to find out

Bottom 5% = 5th percentile = [tex]4.98-1.645(0.05) \\= 4.89775[/tex]

Top 5% = 95th percentile = [tex]4.98+1.645(0.05) \\= 5.06225[/tex]

i.e. we get the items to be qualified as per required measurements would fall between 4.90 and 5.06 cm

If any nail has length less than 4.90 cm or more than 5.06 cm the item would be rejected.