This document discusses hypothesis testing and the different types of errors that can occur. It defines the null hypothesis as the initial assumption being tested, and the alternative hypothesis as the statement the test is aiming to establish. A Type I error occurs when the null hypothesis is falsely rejected, while a Type II error happens when a false null hypothesis is not rejected. Examples are given of testing the hypotheses that adding water or fluoride to toothpaste protects against cavities, and the potential type I and II errors in each case.

1. HYPOTHESIS
TYPE I AND TYPE II ERROR
NEETHU SUSAN ABRAHAM
ROLL NO: 23

2. HYPOTHESIS TESTING
Hypothesis testing refers to
– Making an assumption, called hypothesis, about a
population parameter.
– Collecting sample data.
– Calculating a sample statistic.
– Using the sample statistic to evaluate the hypothesis
(how likely is it that our hypothesized parameter is
correct. To test the validity of our assumption we
determine the difference between the hypothesized
parameter value and the sample value.)

3. NULL HYPOTHESIS
The null hypothesis H0 represents a theory that has been
put forward either because it is believed to be true or
because it is used as a basis for an argument and has not
been proven. For example, in a clinical trial of a new drug,
the null hypothesis might be that the new drug is no better,
on average, than the current drug. We would write H0:
there is no difference between the two drugs on an
average.

4. ALTERNATIVE HYPOTHESIS
The alternative hypothesis, HA, is a statement of what a
statistical hypothesis test is set up to establish. For
example, in the clinical trial of a new drug, the alternative
hypothesis might be that the new drug has a different
effect, on average, compared to that of the current drug.
We would write
HA: the two drugs have different effects, on average. or
HA: the new drug is better than the current drug, on
average.

5. TYPE I AND TYPE II ERROR
Type I error refers to the situation when we reject the null hypothesis
when it is true (H0 is wrongly rejected).
• e.g. H0: there is no difference between the two drugs on average.
• Type I error will occur if we conclude that the two drugs produce
different effects when actually there isn’t a difference.
• Prob(Type I error) = significance level = α
Type II error refers to the situation when we accept the null
hypothesis when it is false.
• H0: there is no difference between the two drugs on average.
• Type II error will occur if we conclude that the two drugs produce
the same effect when actually there is a difference.
• Prob(Type II error) = ß

6. EXAMPLE 1
– Hypothesis: "Adding water to toothpaste protects
against cavities."
– Null hypothesis: "Adding water to toothpaste has no
effect on cavities."
– This null hypothesis is tested against experimental data
with a view to nullifying it with evidence to the contrary.
– A type I occurs when detecting an effect (adding water to
toothpaste protects against cavities) that is not present.
The null hypothesis is true (i.e., it is true that adding
water to toothpaste has no effect on cavities), but this
null hypothesis is rejected based on bad experimental
data.

7. EXAMPLE 2
– Hypothesis: "Adding fluoride to toothpaste protects
against cavities."
– Null hypothesis: "Adding fluoride to toothpaste has no
effect on cavities."
– This null hypothesis is tested against experimental data
with a view to nullifying it with evidence to the contrary.
– A type II error occurs when failing to detect an effect
(adding fluoride to toothpaste protects against cavities)
that is present. The null hypothesis is false (i.e., adding
fluoride is actually effective against cavities), but the
experimental data is such that the null hypothesis
cannot be rejected.