Summary of Hypothesis Testing

Step 1: Setup the alternative (research hypothesis, H_a, and set H₀: m = m ₀)

Hypothesis

Left-sided

Right-sided

Null vs. Alternative

H₀: m = m ₀, H_a: m < m ₀

H₀: m = m ₀, H_a: m ¹ m ₀

H₀: m = m ₀, H_a: m > m ₀

Step 2: Setup the appropriate rule and test statistic

Summary of Hypothesis Testing

Carl Lee

A. Procedure for Testing a Hypothesis mean -Large Sample Cases

1. The following table summarizes the procedure.

Step 1: Setup the alternative (research hypothesis, H_a, and set H₀: m = m ₀)
Hypothesis	Left-sided	Two-sided	Right-sided
Null vs. Alternative	H₀: m = m ₀, H_a: m < m ₀	H₀: m = m ₀, H_a: m ¹ m ₀	H₀: m = m ₀, H_a: m > m ₀
Step 2: Setup the appropriate rule and test statistic

Rule: use z-statistic

Step 3: Compute the observed z-value using observed sample mean and s.d., s from the collected sample data:
Reject H₀ (in favor of H_a)	z_obs < -za	z_obs < -za or z_obs > za	z_obs > za
Step 4: Make a concluding statement based on the question asked.

B. Note, in all hypothesis testing situations, the direction of the alternative hypothesis determines the direction of the rejection region. For example, in the large-sample test of hypothesis about a population mean, for H_a: m > m ₀, the rejection region is z > za ; for H_a: m < m ₀, the rejection region is z < -za ; for H_a: m ¹ m ₀, the rejection region is z < -za _/2 or z > za _/2. Note also, that, whenever H_ais two-tailed, a is divided into 2 equal parts to determine the rejection region.

C. p-value and a-value:

The a-value is the predetermined level of significance. Typical a-value is 1%, 2%, 5% or 10%. This value sets up the rejection region. The reason that a-value is usually small is because , in real applications, we should not reject null hypothesis unless we have a strong evidence that the sample mean is far away from the null hypothesis. Setting a-value as small as 5% indicates that if the sample mean really falls into the 5% region, then, it must be very rare compared with the null, and hence, we should strong evidence to reject the null hypothesis.

On the contrast, p-value is the observed level of significance. One can compare the p-value with the a -value to make the decision in hypothesis test problems. This is a common approach when computer software is available. It is not easy to be computed by hand. However, all statistical software gives the p-value for the intended test.

D. The difference between p-value and a -value. Consider a right-sided test situation,

a -value = P(Z > z_a ). This is the probability of Z higher than the critical value z_a .

p-value = P(Z > z_obs) This is the probability of Z higher than the observed z_obs. For example, consider a = .05. Then z_a = 1.645, and suppose we obtain z_obs = 1.02

Based on the rule using z_a and z_obs, we see, since z_obs = 1.02 < z_a = 1.645, we do not reject H₀. If we compute the

p-value, P(Z > z_obs) = P(Z > z_1.02) = 0.5 - .3461 = .1539. Then, in stead of comparing the two z-values (z_obs vs. za ), we can compare the corresponding "rejection region" probabilities (p and a ).

E. The rule based on the p-value will be:

If the p-value < a , then, reject H₀, otherwise do not reject H₀.

Based on this rule, p-value = .1539 > a = .05. We do not reject H₀.

Note: the final decision using either (z_obs, za ) or (p-value, a ) are the same.

F. Computation of the p-value and the corresponding rule-large sample case

Left-sided test: p-value = P(Z < z_obs)

Two-sided test: p-value = 2P(Z > |z_obs|)

Right-sided test: p-value = P(Z > z_obs)

G. The rule for using the p-value, regardless of the type of test is

If p-value < a , the reject H₀

If p-value ³ a , then do not reject H₀

H. Procedure for Testing a Hypothesis mean-Small Sample Cases

1. The following table summarizes the procedure.

Rule: use t-statistic

Step 3: Compute the observed t-value using observed sample mean and s.d., s from the collected sample data:
Reject H₀ (in favor of H_a) if	t_obs < -t₍a ,n )	t_obs < -t₍a /2, n ) or t_obs > t₍a /2, n )	t_obs > t₍a , n )
Or use p-value to make a decision
	p-value = P(t < t_obs)	p-value = 2P(t > \|t_obs\|)		p-value = P(t > t_obs
Decision based on p-value If p-value < a then reject H₀. If p-value ³ a then do not reject H₀.
Step 4: Make a concluding statement based on the question asked.

2. The difference between small sample and large sample hypothesis testing problems is the choice of test statistics. A test statistic is a standardization of the sample statistic. For large sample cases, the standardized is . For small sample cases, the standardized is even though the computation is the same.

I. The concept that a test-statistic is the standardized sample statistic is true for most of statistical hypothesis testing problems, including all test statistics that are/will be covered in this course and many others that are not covered I this course. The reasons behind using the standardized sample statistics as test statistics is that standardized measures can be compared without worrying about the units, and probability distributions for the standardized sample statistics are either known or easier to derive.

J. Procedure for Testing a Hypothesis on Proportion.

1. Procedure for testing a hypothesis on population proportion is similar to the large sample test for mean, except that we are interested in p (population proportion) not in m (population m ).

Rule: use z-statistic

Step 3: Compute the observed z-value using the sample proportion :

Reject H₀ (in favor of H_a)

z_obs < -za

z_obs < -za or z_obs > za

z_obs > za

Step 4: Make a concluding statement based on the question asked.

2. In step 3 of the procedure, Z is the standardized variable for through where