Answer :
Answer: 0.2551
Step-by-step explanation:
Given : The temperature reading from a thermocouple placed in a constant-temperature medium is normally distributed with mean μ, the actual temperature of the medium, and standard deviation σ.
Significance level : [tex]\alpha=1-0.95=0.05[/tex]
The critical z-value for 95% confidence : [tex]z_{\alpha/2}=1.960[/tex] (1)
Since , [tex]z=\dfrac{x-\mu}{\sigma}[/tex] (where x be any random variable that represents the temperature reading from a thermocouple.)
Then, from (1)
[tex]\dfrac{x-\mu}{\sigma}=1.96\\\\ x-\mu=1.96\sigma[/tex] (2)
Also, all readings are within 0.5° of μ,
i.e. [tex]x-\mu<0.5[/tex]
i.e. [tex]1.96\sigma<0.5[/tex] [From (2)]
i.e. [tex]\sigma<\dfrac{0.5}{1.96}=0.255102040816[/tex]
i.e. [tex]\sigma\approx0.2551[/tex]
The required standard deviation : [tex]\sigma=0.2551[/tex]