Question

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. What would the value of σ have to be to ensure that 95% of all readings are within 0.5° of μ? (Round your answer to four decimal places.) σ =

Answer 1

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 : $\alpha=1-0.95=0.05$

The critical z-value for 95% confidence : $z_{\alpha/2}=1.960$ (1)

Since , $z=\dfrac{x-\mu}{\sigma}$ (where x be any random variable that represents the temperature reading from a thermocouple.)

Then, from (1)

$\dfrac{x-\mu}{\sigma}=1.96\\\\ x-\mu=1.96\sigma$     (2)

Also,  all readings are within 0.5° of μ,

i.e. $x-\mu$

i.e. $1.96\sigma$   [From (2)]

i.e. $\sigma$  

i.e. $\sigma\approx0.2551$

The required standard deviation : $\sigma=0.2551$