Type I and Type II Error Calculation Lecture 33

 Type I & Type II Error

Lecture 33

Determination of Critical Value “c”

The critical value is a threshold value that divides the distribution of a test statistic graph or curve into acceptance and rejection regions. The null hypothesis is rejected if the test statistic value lies within the rejection region; if not, it cannot be rejected. The critical value depends on alternative hypotheses, whether one-tailed or two-tailed. A two-tailed test will have two critical regions, and a one-tailed test will have just one critical region.

 

* when the alternative hypothesis of the form:

H0: θ ≠ θ0

Then c value can obtained as;

* when the alternative hypothesis of the form:

H0: θ > θ0

Then c value can obtained as;

* when the alternative hypothesis of the form:

H0: θ < θ0

Then c value can obtained as;

Inferential Error

In hypothesis testing, the sample data either confirms or refutes the null hypothesis. It is possible to make errors in this situation, such as rejecting a valid null hypothesis or accepting a false null hypothesis or incorrectly. Inferential error is the term for such errors.

1. Type Error

If an investigator rejects a true null hypothesis, then it is called a type I error. The probability of type I error is represented by α.

$$

α =P(RejectH0 / H0 is true)

Examples:

i. A medicen is effective for the treatment of a disease and the doctor does not advise it.

ii. A helmet is important in bike driving, but the biker refuses the helmet. 

iii. A judge punishes an innocent person.

Main Causes of Type I Error

1. When a variable is influenced by a factor other than the variable under test. The outcome of this effect-causing element gives support to the rejection of the null hypothesis.

2. The significance level is established before the hypothesis test.

3. The careless establishment of the null hypothesis.

4. The small sample size and the null hypothesis are disregarded.

Type II Error

If an investigator accepts a false null hypothesis, then it is called a type II error. The probability of type II error is represented by β. 

                                                         β = P(Accept H0 /H0 is False)

                                                        β = P(Accept H0 / H1 is true)

Examples:

i. A medicen is not effective for the treatment of a disease, and the doctor advises it.

ii. A judge releases a guilty person.

iii. A teacher promotes a weak student to the next class.

Main Causes of Type II Error

1. The small sample size.

2. The selection of biased hypotheses.

3. The selection of poor sampling distribution.

Example 8.18: Given H0: μ ≥ 200 vs. μ < 200, n = 100, standard deviation = 25 and α = 0.05

1. For what values of the sample mean H0 be accepted?

2. Compute beta if u is actually 191.

3. Compute the power test, and what does it mean?

Solution: The null hypothesis will be accepted if the sample mean is less than or equal to the critical value.

The null hypothesis will be accepted if the sample mean is less than or equal to 195.8875; otherwise, it will be rejected.

It means a 97% chance of a false null hypothesis.

Example 8.19: A random sample of size n = 25 selected from a normal population with a standard deviation of 10. The null and alternative hypotheses are given below:

H0 : μ =170 vs. H1: μ >170

i. The null hypothesis is rejected if the sample mean is equal to or more than 172. Calculate the probability type I error.

ii. Compute the probability type II error if the true population is 173.

iii. Compute the power of the test if the sample mean is more than 172 and the true population mean is 173.

Solution:

i. The probability of type I error is given by;

α = P(Reject H0 / H0 is true)

α = P(X¯ ≥172 / H0 : μ = 170)

ii. The probability of type II error is given by:

β =P(Accept H0 / H1 is true)

β =P(X¯ < 172 / H1 : μ 173)

iii. Power of the test

π =1 - β

π =1 - 0.3085

π = 0.6915

How to reduce Type I & Type II Errors

 

Several strategies can be employed to reduce type I and type II errors.

*Type I errors can be reduced by selecting a lower level of significance at the start of the study. By setting the lower significance threshold, the probability of incorrectly rejecting a null hypothesis decreases.

* Increased sample size also reduces the probability of type I error and type II error.

 

The Power of a Test

The power of a test is the probability of making the correct decision if the alternative hypothesis is true. Or we can say the probability of rejecting a false null hypothesis. The power of a test is the complement of type II error.

π =1 - P(Reject H0 /H0 is False)

π =1 - P(Reject H0 /H1 is True)

π =1 - β

The power curve can be obtained by plotting parametric values under alternative hypotheses on the x axis and π values on the y axis and joints by line segments, which gives an "S" type curve.

Operating chrematistic Curve

(OC Curve)

The operating characteristic curve is the visual representation of the probability of type II error. To construct OC curve parametric values under alternative along x axis and probability of type II along y axis and gave a reverse “S” shaped curve. The OC curve is the complement of the power curve.

Example 8.20: A random sample of size 4 is selected from a normal population with a known standard deviation of 3.873. a hypothesis of the form u = 30 against u > 30 at 5% significance level. Calculate the following:

i. The critical value of the sample mean at which the null hypothesis is rejected.

ii. Type II error for values of u = 31, 32, 34, 35, 36, 37, 38 in the alternative hypothesis.

Power of the test and sketch power curve and OC curve.

Solution:

ii. Probability of type II errors:

Power of the test:

• Read More: Introduction to chi Square distribution