Math Snap

STEP 1

1. We are using the Student's t-distribution to find the P-value.
2. The test statistic $t$ and sample size $n$ are given.
3. The alternate hypothesis $H_1$ is specified for each part.
4. We will use a t-distribution table to find the range for the P-value.

STEP 2

1. Identify the degrees of freedom.
2. Use the t-distribution table to find the P-value range for part (a).
3. Use the t-distribution table to find the P-value range for part (b).

STEP 3

Calculate the degrees of freedom for part (a) and part (b).
For part (a), $n = 21$, so degrees of freedom $df = n - 1 = 20$.
For part (b), $n = 15$, so degrees of freedom $df = n - 1 = 14$.

STEP 4

For part (a), $t = -1.498$, $df = 20$, and $H_1: \mu \neq \mu_0$.
Look up the critical values for $t = -1.498$ with $df = 20$ in the t-distribution table.
The P-value is between 0.10 and 0.20.

For part (b), $t = -1.585$, $df = 14$, and $H_1: \mu < \mu_0$.
Look up the critical values for $t = -1.585$ with $df = 14$ in the t-distribution table.
The P-value is between 0.10 and 0.15.
The P-value ranges are:
- Part (a): $0.10 < P \text{-value} < 0.20$
- Part (b): $0.10 < P \text{-value} < 0.15$