Let's break down the solution step-by-step:
1. Identify the parameters:
- Significance level [tex]\(\alpha = 0.1\)[/tex].
- Test statistic [tex]\(z = -1.71\)[/tex].
2. Finding the P-value:
The P-value is the probability that the observed statistic is at least as extreme as the test statistic, given that the null hypothesis is true. Since we are using a [tex]\(z\)[/tex]-test, we will use the cumulative distribution function (CDF) of the standard normal distribution.
- For a [tex]\(z\)[/tex]-value of [tex]\(-1.71\)[/tex], we need to find the cumulative probability to the left of [tex]\(z = -1.71\)[/tex] in the standard normal distribution.
The P-value rounded to four decimal places is:
[tex]\[
\boxed{0.0436}
\][/tex]
3. Finding the critical values:
Since we are given a significance level [tex]\(\alpha = 0.1\)[/tex] for a two-tailed test, we will split this [tex]\(\alpha\)[/tex] into two tails, with each tail getting [tex]\(\alpha/2 = 0.05\)[/tex].
- We need to find the z-scores that correspond to the cumulative probabilities of [tex]\(0.05\)[/tex] and [tex]\(1 - 0.05 = 0.95\)[/tex] in the standard normal distribution.
The critical values, rounded to two decimal places, are:
[tex]\[
\boxed{-1.64, 1.64}
\][/tex]
In conclusion:
- The P-value is [tex]\( \boxed{0.0436} \)[/tex].
- The critical values are [tex]\( \boxed{-1.64, 1.64} \)[/tex].
These computations align with the necessary steps for understanding hypothesis testing using the normal distribution.
