What are types of error?
‘To err is human’ is a very famous quote. But do numbers lie? Or can they be mistaken for a fact, which is in fact not true? What will be the consequences if an error has been made? Will there be any adverse effects or can the errors be minimized at the initial stage itself? Are some errors more dangerous than others? These and many such questions are answered in this article on Types of Errors.
What are Type 1 and Type 2 errors?
In statistics, when the hypothesis of testing is done, a null and an alternative hypothesis is set up . A null hypothesis is a hypothesis of no difference and is set up with the intent of it being rejected. In its course, a couple of errors , Type 1 error and Type2 error , are the different types of experimental errors , that might occur. A type 1 error is the probability of the null hypothesis being rejected, when actually it is true. It is also called producer’s risk. It represents a false positive. A type 2 error, on the other hand, is the probability of the null hypothesis being accepted, when in fact it is false. It is called consumer’s risk and represents false negative. Apart from these errors, another measure is the power of the test, which is the probability of the null hypothesis being rejected when it is actually false.
Notation wise, the following holds true:
Type 1 error, α = P(Reject H0|H0)
Type 2 error, Β = P(Accept H0|H1)
Power of the test = 1-P(Accept H0|H1)
= P(Reject H0|H1)
Type 1 error generally means that a person with a certain type of disease has been tested positive for it when actually he doesn’t have the disease and type 2 error means that a person who has the disease has been tested negative. In normal circumstances, type 1 error is considered more dangerous than the other. Since it is more dangerous , it is better to take actions in order to minimize making such errors. This gives rise to the concept of level of significance in the testing of hypothesis. A set alpha is considered in such a way that the type 1 error is at the most the value of alpha and then beta is played around with. Calculating the type of error is a mere calculation of probabilities, some conditional and some unconditional.
The following image gives an idea about how the aforementioned things look like when they are plotted. To minimize type 1 error, the area cut by the yellow curve on the purple curve has to be further reduced.
Both the graphs are bell shaped, indicating a normal distribution as an underlying assumption, as it is for most of the tests of hypothesis for testing mean.
Our experts have answered the following questions on Types of Error and Experimental Errors:
Q. It has been observed many times that in the case of a certain memory test, recognition is much better than recall. However, the probability value in the case of the data from your sample was .12; consequently, you could not reject the null hypothesis that recall and recognition deliver the same results. What kind of error did you end up making?
Q. You have an SRS of size n = 9 from a Normal distribution with Sigma = 1. You wish to test H0: u=0 (u is mu) Ha: u > 0. You decide to reject Ho if x-bar > 0 and accept Ho otherwise.
A) Find the probability of a Type 1 error i.e. find the probabilitity that the test rejects Ho when in fact u = 0.
B).Find the probablity of a Type 2 error when u=.3. This is the probability that the test accepts H0 when in fact u =.3
C). Find the probability of a Type 2 error when u=1.
Q. Let X be binomial random variable with
For the following hypothesis test
we reject as X = 0 or 3.
α = The probability of type 1 error
β = The probability of type 2 error.
|Q. Every day that I pick up lunch down in the Sierra Center I end up spending about $7 give or take about $.50 or so.
a. Draw a rough sketch of this distribution of my lunch expenditures.
b. If, without looking at the total, I pay for my lunch with a $10 bill and receive $1 back in change, should I worry that I was overcharged? Explain your answer in terms of probability.
c. What would be a Type 1 error for this example?
d. What would be a Type 2 error for this example?
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