The **observed significance level, **or
**P-value,**
for a specific statistical test is the probability (assuming the null hypothesis
is true) of observing a value of the test statistic that is at least as
contradictory to the null hypothesis, and supportive of the alternative
hypothesis as the actual one computed from the sample data.

**Decision Criterion for a Hypothesis Test
Using the P-value:**

If P-value is less than a, reject the null hypothesis; otherwise, fail to reject the null hypothesis.

**Examples:**

H_{a}: µ
30 versus H_{o}: µ = 30

Assumptions: X is normally distributed
with s
= 8

Test Statistic:

a = .05 RR: z <
-1.96 or z >1.96 (P-value < .05)

Calculation: z = 1.54

P-value = 2P(z > |z_{calculated}|)
= 2P(z > |1.54|) = 2P(z < -1.54)

= 2(.0618) = .1236

Decision: Fail to reject H_{o}.

**Suppose s is
not known and n = 31.**

**Test Statistic: **

a = .05
RR: t < -2.042 or t > 2.042 (P-value < .05)

**Calculation:** t = 1.54 df = 30

P-value = 2P(t > |t_{calculated }|)
= 2P(t > 1.54)

P(t > 1.310) = .10 and P(t > 1.697) = .05.

Hence, .05 < P(t > 1.54) < .10; therefore .10 < P-value < .20.

**Decision:** Fail to reject H_{o}.

**P-value Calculations**

I. For each test of hypothesis, compute the
p-value. Sketch a figure.

1) H_{a}: µ
< 100 versus H_{o}: µ __>__ 100

If n = 20 and t = -1.78, find the p-value of
t = -1.78.

2) H_{a}: µ
> 240 versus H_{o}: µ __<__ 240

If z = 2.35, find the p-value of z = 2.35.

3) H_{a}: µ 75
versus H_{o}: µ = 75

If n = 22 and t = -2.236, find the p-value
of t = -2.236.

4) H_{a}:
µ > 80 versus H_{o}: µ __<__ 80

If t = 2.35 and n = 24, find the p-value of
t = 2.35.

5) H_{a}: s_{1}^{2}
< s_{2}^{2}
versus H_{o}: s_{1}^{2}
__>__
s_{2}^{2}

If F = 4.63, n_{1} = 18 and n_{2}
= 16, find the p-value of F = 4.63.

6) H_{a}: s^{2}
< 144 versus H_{o}: s^{2}
__>__
144

If c_{n-1}^{2}
= 7.44 and n = 18, find the p-value of c_{n-1}^{2}
= 7.44.