The temperature of a certain solution is estimated by taking a large number of independent measurements and averaging them. The estimate is 37°C, and the uncertainty (standard deviation) in this estimate is 0.1°C.

a. Find a 95% confidence interval for the temperature.

b. What is the confidence level of the interval 37 ± 0.1°C?

c. If only a small number of independent measurements had been made, what additional assumption would be necessary in order to compute a confidence interval?

d. Making the additional assumption, compute a 95% confidence interval for the temperature if 10 measurements were made.

Answer:

Step 1 of 4:

(a)

In this question, we are asked to find a confidence interval for the temperature.

Given the temperature of a certain solution is estimated by taking a large number of independent measurements and averaging them.

The estimate is , and the uncertainty (standard deviation) in this estimate is .

Hence our sample mean is and standard deviation

Since the large number of independent measurements has been taken, then a level confidence interval for temperature is,

………(1)

we want 95% confidence interval, then a level is

and

Then for is , hence

A 95% confidence interval for temperature is,

Hence the 95% confidence interval for temperature is (36.804, 37.196).

Step 2 of 4:

(b)

In this question, we are asked to find the confidence level of the interval .

We know the confidence level of the interval can be written like this,

…………..(2)

We have given the interval,

………..(3)

Compare equation (2) and (3), we have,

Using the , the area to the left of is approximately .

Therefore , so the level is,

or approximately .

Hence the confidence level of the interval is 68%.