Question

You perform a hypothesis test on the claim that the volume in all 12-ounce cans of Fizzy Pop is less than 12 ounces. In terms of the possible conclusions of this hypothesis test, choose the appropriate description of a Type I and Type II error.

(a) If you make a Type I error, then which of the following is true?

You conclude that the mean volume is 12 ounces when it isn't.

You conclude the mean volume in all cans is less than 12 ounces when that is not true.     

You don't conclude the mean volume in all cans is less than 12 ounces when in fact, the mean volume is less than 12 ounces.

(b) A Type II error occurs when which of the following is true?

You conclude the mean volume in all cans is less than 12 ounces when that is not true.

You don't conclude the mean volume in all cans is less than 12 ounces when in fact, the mean volume is less than 12 ounces.     

You conclude that the mean volume is 12 ounces when it isn't.

(c) What is the probability of a Type I error if the significance level is α?

1 − α

1 − 2α    

α

2α

Answer 1

According to the problem :-

Null hypothesis : Mean volume in all 12-ounce cans is at least 12 ounces.

Alternative hypothesis : Mean volume in all 12-ounce cans is less than 12 ounces.

.

(a)

Type I error occurs when null hypothesis is true but is rejected in accordance to the sample obtained. Thus if you make a Type I error, you conclude the mean volume in all cans is less than 12 ounces when that is not true.

.

(b)

Type II error occurs when null hypothesis is false but is accepted in accordance to the sample obtained. Thus if you make a Type II error, you don't conclude the mean volume in all cans is less than 12 ounces when in fact, the mean volume is less than 12 ounces.

.

(c)

Type I error is considered to be more serious than Type II error. So the probability of Type I error is fixed and subject to this condition, the test with minimum probability of Type II error is chosen. The probability of Type I error is therefore taken as the level of significance of a test. Thus if the significance level is , the probability of Type I error is also .