Means of two normal populations with known variances

Suppose that the random variable is based on a random sample of

size from a normal population with mean and known variance .

Also suppose that the random variable is based on a random sample of

size from a normal population with mean and known variance .

The difference between population means has a mean and variance .

Therefore, the random variable

has a standard normal distribution.

We can use this fact to obtain confidence interval for the difference between the population means.

Definition:

When the variances and of two normal are known, then

confidence interval for is given by

.

Example:

A sample of size 13 from a normal population with variance 100

yielded . A sample of size 7 from a second normal population with variance 80 yielded .Find a 95 % confidence interval for .

Solution:

and

The 95 % confidence interval for is

Remark:

When and are both large, the normal approximation remains valid if and are replaced by their estimators and . When and are greater than 30, an approximate confidence interval for

is given by

where is the number for which

and Z follows standard normal distribution.

Example:

A sample of 50 yogurt cups produced by the company showed that they contain an average of 146 calories per cup with a standard deviation of

6.4 calories. A sample of 60 such yogurt cups produced by its competitor showed that they contained an average of 143 calories per cup with a standard deviation of 7.2 calories. Make a 97 % confidence interval for the difference between the mean number of calories in yogurt cups produced by the two companies.

Solution:

We can refer to the respective samples as sample 1 and sample 2.

Let and be the means of populations 1 and 2 respectively, and

let and be the means of the respective samples.

From the given information:

; ;

; ;

Since both sample sizes are large (, ) we can replace and by and respectively.

Then confidence interval for is given by

and

Finally, substituting all the values in the confidence interval formula, we obtain 97 % confidence interval for as

.

Thus, with 97 % confidence we can state that the difference in the mean calories of the two population of yogurt cups produced by two different companies is between 0.18 and 5.82.

Дата добавления: 2015-08-05