Question

A sample of 64 account balances from a credit company showed an average daily balance of $1,040.
The standard deviation of the population is known to be $200. We are interested in determining if the
mean of all account balances (i.e., population mean) is significantly different from $1,000.What is the
probability of making a of the Type II error if the mean is 945?

Answer 1

As we are testing here whether the mean is different from 1000, therefore this is a two tailed test. The test statistic here is computed as:

Now as this is a two tailed test, the p-value here is computed as:

p = 2P(Z > 1.6 )

Getting this from the standard normal tables, we get:

p = 2P(Z > 1.6 ) = 2*0.0548 = 0.1096

As the p-value here is very large = 0.1096 > 0.05 which could be taken the level of significance, therefore the test is insignificant and we cannot reject the null hypothesis here and therefore we dont have sufficient evidence here that the mean is different significantly from 1000.

The type II error probability is the probability of retaining the null hypothesis given that it is false.

Therefore given that the actual mean is 945, probability that it is retained is computed as:

Note that we are not given the level of significance here.

From standard normal tables, P( -1.96 < Z < 1.96 ) = 0.95

Therefore, it wont be rejected if:

Converting this to a standard normal tables, we get:

Therefore 0.4052 is the required probability here.