Confidence interval for : small samples

Let us turn our attention to finding confidence interval for when sample size is small. Using

We can derive the formula for confidence interval for the case when a small sample is selected from a normally distributed population with mean and unknown variance. It is given by

where is the number for which

The random variable has a Student’s t distribution with degrees of freedom. (Fig. 6.10).

Remark:

If the sample is available, then standard deviation can be calculated as

, where or

Example:

For the t distribution with n =10, find the number b such that

Solution:

The probability in the interval

is 0.80. (Fig. 6.11).

We must have a probability of 0.10 to the right of b and a probability of 0.10 to the left

of – b.

So

and .

Example:

A random sample of 25 busses shows a sample mean of 225 passengers carried per day per bus. The sample standard deviation is computed to be 60 passengers. Find a 90% confidence interval for the mean number of passengers carried per bus during a 1 –day period.

Solution:

A 90 % confidence interval for the mean is given by

, so

=90 %

and

After substitution we obtain

or .

We are 90 % confident the mean number of passengers carried per day by bus is between 204.5 and 245.5, because 90 % of the intervals calculated in this manner will contain the true mean number of passengers carried per day per bus.

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