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

Answer:

15.9% (nearest tenth)

Step-by-step 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:

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

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

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