Answer:
The value is [tex]p-value = 0.0297[/tex]
Step-by-step explanation:
From the question we are told that
The population mean is [tex]\mu = 420 \ g[/tex]
The The sample size is [tex]n = 49[/tex]
The sample mean is [tex]\= x = 413 \ g[/tex]
The population variance is [tex]\sigma^2 = 676[/tex]
Generally the population standard deviation is mathematically represented as
[tex]\sigma = \sqrt{\sigma^2 }[/tex]
[tex]\sigma = \sqrt{676}[/tex]
[tex]\sigma = 26[/tex]
The null hypothesis is [tex]H_o : \mu = 420 \ g[/tex]
The alternative hypothesis is [tex]H_a : \mu < 420 \ g[/tex]
Generally the test statistics is mathematically represented as
[tex]z = \frac{ \= x - \mu }{ \frac{ \sigma }{\sqrt{n} } }[/tex]
=> [tex]z= \frac{ 413 - 420 }{ \frac{ 26 }{\sqrt{49} } }[/tex]
=> [tex]z = -1.884[/tex]
The p-value is obtained from the z-table table and the value is
[tex]p-value = P(Z < -1.885) = 0.029715[/tex]
[tex]p-value = 0.0297[/tex]
