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

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=(x-\mu)/(\sigma)` (where x be any random variable that represents the temperature reading from a thermocouple.)

Then, from (1)

`(x-\mu)/(\sigma)=1.96\\\\ x-\mu=1.96\sigma` (2)

Also, all readings are within 0.5° of μ,

i.e.
`x-\mu<0.5`

i.e.
`1.96\sigma<0.5` [From (2)]

i.e.
`\sigma<(0.5)/(1.96)=0.255102040816`

i.e.
`\sigma\approx0.2551`

The required standard deviation :
`\sigma=0.2551`