In: Math
a. What is hypothesis testing in statistics? Discuss
b. Does Type I error being considered more serious than Type II error? Explain
c. What is the p-value of a test? Give an example
Answers of A,B,C
A)
Hypothesis testing is a statistical method that is used in making
statistical decisions using experimental data. Hypothesis Testing
is basically an assumption that we make about the population
parameter.Hypothesis testing was introduced by Ronald Fisher, Jerzy
Neyman, Karl Pearson and Pearson’s son, Egon
Pearson.
The main purpose of statistics is to test a hypothesis. For
example, you might run an experiment and find that a certain drug
is effective at treating headaches. But if you can’t repeat that
experiment, no one will take your results seriously. A good example
of this was the cold fusion discovery, which petered into obscurity
because no one was able to duplicate the results
Hypothesis testing is an act in statistics whereby an analyst tests
an assumption regarding a population parameter. The methodology
employed by the analyst depends on the nature of the data used and
the reason for the analysis. Hypothesis testing is used to infer
the result of a hypothesis performed on sample data from a larger
population.
Key terms and concepts:
Null hypothesis: Null hypothesis is a statistical hypothesis that
assumes that the observation is due to a chance factor. Null
hypothesis is denoted by; H0: μ1 = μ2, which shows that there is no
difference between the two population means.
Alternative hypothesis: Contrary to the null hypothesis, the
alternative hypothesis shows that observations are the result of a
real effect.
Level of significance: Refers to the degree of significance in
which we accept or reject the null-hypothesis. 100% accuracy is not
possible for accepting or rejecting a hypothesis, so we therefore
select a level of significance that is usually 5%.
Type I error: When we reject the null hypothesis, although that
hypothesis was true. Type I error is denoted by alpha. In
hypothesis testing, the normal curve that shows the critical region
is called the alpha region.
Type II errors: When we accept the null hypothesis but it is false.
Type II errors are denoted by beta. In Hypothesis testing, the
normal curve that shows the acceptance region is called the beta
region.
Power: Usually known as the probability of correctly accepting the
null hypothesis. 1-beta is called power of the analysis.
One-tailed test: When the given statistical hypothesis is one value
like H0: μ1 = μ2, it is called the one-tailed test.
Two-tailed test: When the given statistics hypothesis assumes a
less than or greater than value, it is called the two-tailed
test.
Statistical decision for hypothesis testing:
In statistical analysis, we have to make decisions about the
hypothesis. These decisions include deciding if we should accept
the null hypothesis or if we should reject the null hypothesis.
Every test in hypothesis testing produces the significance value
for that particular test. In Hypothesis testing, if the
significance value of the test is greater than the predetermined
significance level, then we accept the null hypothesis. If the
significance value is less than the predetermined value, then we
should reject the null hypothesis. For example, if we want to see
the degree of relationship between two stock prices and the
significance value of the correlation coefficient is greater than
the predetermined significance level, then we can accept the null
hypothesis and conclude that there was no relationship between the
two stock prices. However, due to the chance factor, it shows a
relationship between the variables.
B)
A Type I error, on the other hand, is an error in every sense of
the word. A conclusion is drawn that the null hypothesis is false
when, in fact, it is true. Therefore, Type I errors are generally
considered more serious than Type II errors. The probability of a
Type I error (α) is called the significance level and is set by the
experimenter. There is a tradeoff between Type I and Type II
errors. The more an experimenter protects himself or herself
against Type I errors by choosing a low level, the greater the
chance of a Type II error. Requiring very strong evidence to reject
the null hypothesis makes it very unlikely that a true null
hypothesis will be rejected. However, it increases the chance that
a false null hypothesis will not be rejected, thus lowering power.
The Type I error rate is almost always set at .05 or at .01, the
latter being more conservative since it requires stronger evidence
to reject the null hypothesis at the .01 level then at the .05
level.
Explanation:
When considering a test of a null hypothesis H0 ,a type 1 error is
the decision to reject the null (say it is false), when in fact it
is true; while a type 2 error is a decision to accept the null (or
"fail to reject it"), when in fact it is false.
In a courtroom, the null hypothesis is that the defendant is
innocent while the alternative hypothesis is the opposite
conclusion, that the defendant is guilty. Therefore, a type 1 error
in this context is the conclusion that the defendant is guilty when
in fact the person is innocent; and a type 2 error in this context
is the conclusion that the defendant is innocent (or at least that
there's not enough evidence to convict), when in fact the person is
guilty.
Both kinds of errors are "bad", though societies typically think of
the first kind of error as worse. The second kind of error here is
more "dangerous" to the society, as it could result in letting a
violent criminal go free.
In business, either kind of error can be "bad" or "dangerous". For
example, if your company makes cars and you have tried to make some
aspect of a crash test come out better (safer), the null hypothesis
would be that the change makes no improvement to safety (or even
makes safety worse) while the alternative hypothesis would be that
the change does make an improvement to safety. In this context, a
type 1 error would be the mistaken belief that the change has
improved safety, when in fact it has not (so this could lead to
more people dying and perhaps lawsuits). And a type 2 error would
be the mistaken belief that the change has made no improvement when
in fact it has (so this would mean a "missed opportunity" to
improve safety, which could also lead to more people dying than
they would otherwise).
C)
A p value is used in hypothesis testing to help you support or
reject the null hypothesis. The p value is the evidence against a
null hypothesis. The smaller the p-value, the stronger the evidence
that you should reject the null hypothesis.
P values are expressed as decimals although it may be easier to
understand what they are if you convert them to a percentage. For
example, a p value of 0.0254 is 2.54%. This means there is a 2.54%
chance your results could be random (i.e. happened by chance).
That’s pretty tiny. On the other hand, a large p-value of .9(90%)
means your results have a 90% probability of being completely
random and not due to anything in your experiment. Therefore, the
smaller the p-value, the more important (“significant”) your
results.
When you run a hypothesis test, you compare the p value from your
test to the alpha level you selected when you ran the test. Alpha
levels can also be written as percentages.
P-values are the probability of obtaining an effect at least as
extreme as the one in your sample data, assuming the truth of the
null hypothesis.
The p-value is calculated using the sampling distribution of the
test statistic under the null hypothesis, the sample data, and the
type of test being done (lower-tailed test, upper-tailed test, or
two-sided test).
The p-value for:
a lower-tailed test is specified by: p-value = P(TS ts | H0 is
true) = cdf(ts)
an upper-tailed test is specified by: p-value = P(TS ts | H0 is
true) = 1 - cdf(ts)
assuming that the distribution of the test statistic under H0 is
symmetric about 0, a two-sided test is specified by: p-value = 2 *
P(TS |ts| | H0 is true) = 2 * (1 - cdf(|ts|))
Where:
P Probability of an event
TS Test statistic
ts observed value of the test statistic calculated from your
sample
cdf() Cumulative distribution function of the distribution of the
test statistic (TS) under the null hypothesis
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