Question

a. Find the p-value of the following test given that
x = 990, n = 100, and σ = 25.
H0: μ = 1000
H1: μ < 1000
b. Repeat part (a) with σ = 50.
c. Repeat part (a) with σ = 100.
d. Describe what happens to the value of the test statistic
and its p-value when the standard deviation
increases.

Answer 1

$\bar x=990$

$n=100$

$\mu = 1000$

This is a test of a single population mean and the population standard deviation is known and the sample size is greater than 30 so this is a z-test.

$$Z_{test}=\frac{\bar x- \mu}{\frac{\sigma}{\sqrt n}}$$

Then looking up the calculated Z-score in a table of the standard normal distribution cumulative probability we find the corresponding probability or p-value.

a) $\sigma=25$

$Z_{test}=\frac{990-1000}{\frac{25}{\sqrt {100}}}=-4 \implies$ p-value = 0.000032

b) $\sigma=50$

$Z_{test}=\frac{990-1000}{\frac{50}{\sqrt {100}}}=-2 \implies$ p-value = 0.02275

c) $\sigma=100$

$Z_{test}=\frac{990-1000}{\frac{100}{\sqrt {100}}}=-1 \implies$ p-value = 0.158655

d) If the standard deviation increases the value of the test statistic value decreases and the p-value increases.