Q 9.27 - Elementary Statistics

Question

Refer to Exercise 9.19. Explain what each of the following would mean.

(a) Type I error.

(b) Type II error.

(c) Correct decision.

Now suppose that the results of carrying out the hypothesis test lead to the rejection of the null hypothesis. Classify that conclusion by error type or as a correct decision if in fact the mean length of imprisonment for motor-vehicle-theft offenders in Sydney. 

(d) equals the national mean of 16.7 months.

(e) differs from the national mean of 16.7 months.

Step-by-Step Solution

Answer

(a) Rejecting a Null Hypothesis, when it is true.

(b) Rejecting a Null Hypothesis, when $\left(H_0\right)$ is false.

(c) If the true null hypothesis is not rejected or a false null hypothesis is rejected.

(d) Correct Decision.

(e) Type II error.

Step 1. Given Information.

The null hypothesis is,

$H_0:\mu =16.7$ months.

The alternative hypothesis is,

$H_0:\mu \ne 16.7$ months.

Part (a). Type I error

According to the definition of the type I error it is to reject a null hypothesis when it is true. A type I error would occur in fact $H_0:\mu =16.7 months$ true, that is the mean length of imprisonment for motor-vehicle-theft offenders in Australia is $16.7 months$ but the result of the sampling lead to conclude that the mean length of imprisonment for motor-vehicle-theft offenders in Australia is less than $16.7 months$. 

Part (b). Type II error.

According to the definition of the type II error, it is to not reject a null hypothesis when it $\left(H_0\right)$ is false. A type II error would occur in fact $\mu =16.7 months$ is not to be rejected, but the results of the sampling fall to lead to conclude that the mean length of imprisonment for motor-vehicle-theft offenders is in $16.7 months$.