Math Snap

STEP 1

What is this asking?
We need to find the probability of getting a test statistic as extreme as $z = 1.39$ if the null hypothesis is true, and then decide if that's small enough to reject the null hypothesis given our significance level of $\alpha = 0.04$.
Watch out!
Make sure you're looking at the correct tail of the distribution – this is a right-tailed test!

STEP 2

1. Find the p-value.
2. Compare to alpha.

STEP 3

For a right-tailed test, the p-value is the probability of observing a $z$-score greater than or equal to our test statistic, assuming the null hypothesis is true.
In this case, we want $P(Z \ge 1.39)$.

STEP 4

We can use a $z$-table or technology to find $P(Z < 1.39)$.
Let's say we find that $P(Z < 1.39) = 0.9177$.

STEP 5

Since the total probability under the standard normal curve is 1, we can find the probability in the right tail by subtracting the value we just found from 1:
$$P(Z \ge 1.39) = 1 - P(Z < 1.39) = 1 - 0.9177 = \textbf{0.0823}$$ So, our p-value is 0.0823.

STEP 6

Our significance level, $\alpha$, is 0.04.
This is the threshold we've set for deciding whether to reject the null hypothesis.

STEP 7

Our p-value (0.0823) is greater than $\textbf{0.04}$.

STEP 8

Since our p-value is greater than alpha, we fail to reject the null hypothesis.
There's not enough evidence to reject it at the $\alpha = 0.04$ significance level.

P-value = 0.0823.
We fail to reject $H_0$.