Question

The temperature of a solution will be estimated by taking n independent readings and averaging them. Each reading is unbiased, with a standard deviation of σ = 0.5°C. How many readings must be taken so that the probability is 0.90 that the average is within ±0.1◦C of the actual temperature? Round the answer to the next largest whole number.

Answer 1

68 readings.

Step-by-step explanation:

We need to take this problem as a statistic problem where the normal distribution table help us.

We can start considerating that X is the temperature of the solution, then

`0.9 = P(|\bar{x}-\mu|<0.1)`

`0.9 = P(\frac{|\bar{x}-\mu|}{(\sigma)/(√(n))}<(0.1)/((\sigma)/(√(n))))`

`0.9 = P(|Z|<(0.1)/((\sigma)/(√(n))))`

For a confidence level of 90% our
`Z_(critic)` is 1.645

Therefore,

`(0.1)/((\sigma)/(√(n))) = 1.645`

Substituting for
`\sigma = 5` and re-arrange for n, we have that n is equal to

`n=((1.645\sigma)/(0.1))^2`

`n=((1.645)^2(0.5)^2)/(0.1^2)`

`n=67.65`

`n=68`

We need to make 68 readings for have a probability of 90% and our average is within