Question

A supply of hex nuts is produced, and production records indicate a mean mass of 7.8 g with a standard deviation of 0.3 g. Assuming a normal distribution , estimate the percent of hex nuts with mass less than 7.5 g

Answer 1

15.9% (nearest tenth)

Explanation:

`X \sim \sf N(\mu, \sigma^2)`

Given:

• mean =
  `\mu` = 7.8 g
• s.d. =
  `\sigma` = 0.3 g

`X \sim \sf N(7.8, 0.3^2)`

Using a calculator:

`\implies \textsf{P}(X < \sf 7.5)=0.1586552539=15.9\%\:(nearest\:tenth)`

Converting to z-value:

`\textsf{If }\: X \sim\textsf{N}(\mu,\sigma^2)\:\textsf{ then }\: (X-\mu)/(\sigma)=Z, \quad \textsf{where }\: Z \sim \textsf{N}(0,1)`

`\implies \textsf{P}(X < 7.5)=\textsf{P}\left(Z < (7.5-7.8)/(0.3)\right)=\textsf{P}(Z < -1)`

`\implies \textsf{P}(Z < \sf-1)=0.1586552539=15.9\%\:(nearest\:tenth)`