Hypothesis Tests

p-Value Method with small sample, σ known

Sometimes, we like to use different method to make a decision. Instead of finding rejection regions and verifying the test statistic value z falls in the rejection regions, we can use a quantity called p-value to make our decisions. Let’s see how to use this method.

Suppose we have a random sample of    X1, X2, X3, ..., Xn

and suppose we have already calculated value z of test statistic Z

p-Value can be determined as

P-Value = P(Z ≤ z)         if  (1) Ho : μ < μ0

P-Value = P(Z ⩾ z)         if (2) Ho : μ > μ0

P-Value = P(Z ⩾ |z|)      if (3) Ho : μ ≠ μ0

Then, compare p-value with level of significance α

If p-value ≤ level of significance α,

reject the null hypothesis

If p-value > level of significance α,

reject the null hypothesis

Step 1

State the null hypothesis

Ho : μ = μ0

Step 2

State the alternative hypothesis

(1) Ho : μ < μ0

(2) Ho : μ > μ0

(3) Ho : μ ≠ μ0

Step 3

Assign an appropriate value of level of significance α

Common Values: 0.01 (1%), 0.05 (5%), and 0.10 (10%)

Pick One: Here 0.01 is picked in this case, or it is given in the question

Step 4

Determine a suitable test statistic

The parameter under investigation is used as test statistic

The test statistic is and call it z, n is small and population is supposed normal, if it not sure, check with normal plot and boxplot

z = (X bar - μ)/(σ/√n)  for  μ

This test statistic is used to test hypothesis (1), (2), (3) about μ

Step 5

Determine the probability distribution in the test statistic

Since, the sample is small (n < 30), according to the central limit theorem the test statistic “ z = (X bar - μ)/(σ/√n)  for  μ” is distributed as standard normal (mean 0 and standard deviation 1).

Step 6

Calculate p-value by using

Remember z is calculated value, in this case for small value of n, use this value to calculate p-value

z = (X bar - μ)/(σ/√n)  for  μ

P-Value = P(Z ≤ z)                                 if  (1) Ho : μ < μ0

[EXCEL USE NORMSDIST(z)]

P-Value = P(Z ⩾ z) = 1 -  P(Z ≤ z)        if (2) Ho : μ > μ0

P-Value = 2 . P(Z ⩾ |z|)                         if (3) Ho : μ ≠ μ0

Step 7

Make Decision

Compare the p-value to level of significance α, in this case 0.01

If p-value ≤ level of significance α,

reject the null hypothesis

If p-value > level of significance α,

do not reject the null hypothesis

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