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

Answer:

68 readings.

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

$0.9 = P(|\bar{x}-\mu|$

$0.9 = P(\frac{|\bar{x}-\mu|}{\frac{\sigma}{\sqrt{n}}}$

$0.9 = P(|Z|$

For a confidence level of 90% our $Z_{critic}$ is 1.645

Therefore,

$\frac{0.1}{\frac{\sigma}{\sqrt{n}}} = 1.645$

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

$n=(\frac{1.645\sigma}{0.1})^2$

$n=\frac{(1.645)^2(0.5)^2}{0.1^2}$

$n=67.65$

$n=68$

We need to make 68 readings for have a probability of 90% and our average is within $0.1\°\frac$