Answer:
The probability is  [tex]P(X < 12) = 0.99286[/tex]
Step-by-step explanation:
From the question we are told that
    The population mean is [tex]\mu = 12 \ oz[/tex]
     The  standard deviation is  [tex]\sigma = 0.1 \ oz[/tex]
     The sample mean is  [tex]\= x = 12.1 \ oz[/tex]
     The sample size is  n = 6 packs
 Â
The standard error of the mean is mathematically represented as
       [tex]\sigma_{\= x } = \frac{\sigma}{\sqrt{n} }[/tex]
substituting values
      [tex]\sigma_{\= x } = \frac{0.1}{\sqrt{6} }[/tex]
      [tex]\sigma_{\= x } = 0.0408[/tex]
Given that the can’s contents follow a Normal distribution then then  the probability that the mean contents of a six-pack are less than 12 oz is mathematically represented as
     [tex]P(X < 12) = P ( \frac{X - \mu }{ \sigma_{\= x }} < \frac{\= x - \mu }{ \sigma_{\= x }} )[/tex]
Generally  [tex]\frac{X - \mu }{ \sigma_{ \= x }} = Z (The \ standardized \ value \ of \ X )[/tex]
So
     [tex]P(X < 12) = P ( Z < \frac{\= x - \mu }{ \sigma_{\= x }} )[/tex]
substituting values
    [tex]P(X < 12) = P ( Z < \frac{12.2 -12 }{0.0408} )[/tex]
   [tex]P(X < 12) = P ( Z < 2.45 )[/tex]
From the normal distribution table the value of [tex]P ( Z < 2.45 )[/tex] is Â
      [tex]P (Z < 2.45)0.99286[/tex]
=> Â [tex]P(X < 12) = 0.99286[/tex]
