Hypothesis test -- statistics

1) Null hypothesis:
The null hypothesis (denote by H₀ ) is a statement about the value of
a population parameter (such as mean), and it must contain the condition of equality and must be written with the symbol =, ≤, or ≤.


2. Alternative hypothesis
The Alternative hypothesis (denoted by H₁ ) is the statement that must be true if the null hypothesis is false.

Conclusion
a) Fail to reject the null hypothesis
b) Reject the null hypothesis.

Significance level
The probability of rejecting the null hypothesis when it is called
the significance level α , and very common choices are
α = 0.05 and α = 0.01

Example:
A sample of data for body temperature (n=106, x̄=98.20°, s =0.62),
for a 0.05 significance level, test the claim that the mean body temperature of health adults is equal to 98.6°F.



Solution: We have:
H₀ : μ=98.6 (original claim) H₁ : μ≠98.6
As n>30, the central limit theorem indicates that the distribution of sample means can be approximated by a normal distribution
z= (x̄ - μ)/ (σ/√n) = (98.20-98.6)/(0.62/√106) = -6.64
The test is two-sided because a sample mean significantly less than or greater than 98.6 is strong evidence against the null hypothesis that μ=98.6.
α/2 = 0.025, the left critical z value = -1.96, right z critical value = 1.96. For z between -1.96 and 1.96, we fail to reject H₀.
As z = -6.64, we reject the reject H₀ i.e. we reject the hypothesis that the mean body temperature of health adults is equal to 98.6°F.