Probability of error – all about the topic

The probability of error is a term from statistics. She stands for them Probability of making a type I error in a test or the misclassification rate when evaluating a classifier.

The maximum allowable error rate for a statistical test is also called level of significance designated. If the null hypothesis is rejected at this level, it corresponds to the maximum Type I error probability of rejecting the null hypothesis when it is correct. je smaller this probability of error is all the more the probability of a type 2 error is greaterthat the null hypothesis is not rejected even though it is false.

The level of error does not correspond to the p-value calculated when performing a test. It also does not state the probability that a hypothesis is correct.

In a nutshell:

• Probability of type 1 errors in a test
• The maximum permissible error probability is called the significance level
• The smaller, the higher the probability of type 2 errors

How do I calculate the probability of error?

If the following quantities are given:

• X = binomial random variable
• p 0 = hypothesis H 0
• n = sample length
• k = decision rule

Then the following things can be looked for:

• (alpha) = probability of error

Tip! To calculate P(X ≤ k) you usually need a pocket calculator or a statistical table.

Example of task 1- probability of error

Now let’s apply the whole theory with an example.

The task is:

Before a major customer buys a very large number of chocolate bars, the hypothesis – H 0: Less than 10% of the chocolate bars are damaged – tested. The following decision rule has been defined for this purpose:

10 panels are viewed. If 2 or more tablets are found to be faulty, the H 0 hypothesis is rejected and the purchase will not take place.

Determine the probability of a type I error.

1. Write down the given things.
   • H 0: p ≤ 0.1 (So: right-sided hypothesis test with p 0 = 0.1)
   • n = 10 (sample length)
   • k = 2 (decision rule: from k = 2, H 0 is rejected)
2. Write down the things you are looking for.
   • (Probability of error, probability of error of the 1st type)
   • The following applies: = (0 or 1 table are incorrect)

With a probability of 26%, a type I error is committed with the decision rule. The decision rule is not particularly well suited.

Example Task 2 – probability of error

The party executive of the party for more politics (FMP for short) announces the following decision-making rule: «We ask 1,000 randomly selected people whether they will vote for us next Sunday.»

If 160 or more people say that they will vote for us, then I reject my hypothesis that we will get a maximum of 15% of the votes.

1. Suppose 160 or more respondents say they vote for the For More Politics party. Which of the two beliefs will replace the board’s hypothesis?

• A: We will get less than 15% of the votes.
• B: We will get more than 15% of the votes.

2. Calculate the significance level at which the party executive tests its decision rule.

Solution:

1. Since 160 or more of the people surveyed say they vote for the FMP party, an even larger share of the vote is expected. So the belief is:

• B: We will get more than 15% of the votes.

2.

• Given is:
  • (So: right-sided hypothesis test with p= 15%.)
  • (sample length)
  • Decision rule: From k = 160 it is rejected.

• Wanted is:
  • (Probability of error, probability of error of the first kind.)

The following therefore applies:

With a probability from scarce 20% becomes a with this decision rule Error 1st kind committed.

Probability of error – everything important at a glance

As the name suggests, you use the probability of error to calculate how high the probability is that an error will occur. You have to read the assignment very carefully.

Then you calculate the probability of error in the following steps:

1. Write down given and wanted quantities. Determine the decision rule.
2. Write down the error rate in the formula. Be it an error of the 1st or 2nd type, you must decide or infer from the task which error is being discussed.
3. calculate probability. Interpret result and formulate answer.