Sure, let's determine the closest \( z \)-value for each given probability: 0.14, 0.16, 0.86, and 0.98. We have a table of \( z \)-values and their corresponding probabilities:
[tex]\[
\begin{array}{|c|c|}
\hline
z & \text{Probability} \\
\hline
0.00 & 0.5000 \\
\hline
1.00 & 0.8413 \\
\hline
2.00 & 0.9772 \\
\hline
3.00 & 0.9987 \\
\hline
\end{array}
\][/tex]
To determine the closest \( z \)-value for each probability, we compare the given probabilities with the available probabilities in the table.
1. For the probability 0.14:
- Differences:
- \( |0.5000 - 0.14| = 0.3600 \)
- \( |0.8413 - 0.14| = 0.7013 \)
- \( |0.9772 - 0.14| = 0.8372 \)
- \( |0.9987 - 0.14| = 0.8587 \)
- The smallest difference is 0.3600, corresponding to \( z = 0.00 \).
2. For the probability 0.16:
- Differences:
- \( |0.5000 - 0.16| = 0.3400 \)
- \( |0.8413 - 0.16| = 0.6813 \)
- \( |0.9772 - 0.16| = 0.8172 \)
- \( |0.9987 - 0.16| = 0.8387 \)
- The smallest difference is 0.3400, corresponding to \( z = 0.00 \).
3. For the probability 0.86:
- Differences:
- \( |0.5000 - 0.86| = 0.3600 \)
- \( |0.8413 - 0.86| = 0.0187 \)
- \( |0.9772 - 0.86| = 0.1172 \)
- \( |0.9987 - 0.86| = 0.1387 \)
- The smallest difference is 0.0187, corresponding to \( z = 1.00 \).
4. For the probability 0.98:
- Differences:
- \( |0.5000 - 0.98| = 0.4800 \)
- \( |0.8413 - 0.98| = 0.1387 \)
- \( |0.9772 - 0.98| = 0.0028 \)
- \( |0.9987 - 0.98| = 0.0187 \)
- The smallest difference is 0.0028, corresponding to \( z = 2.00 \).
So, the closest \( z \)-values for the given probabilities are:
- Probability \( 0.14 \) -> \( z = 0.00 \)
- Probability \( 0.16 \) -> \( z = 0.00 \)
- Probability \( 0.86 \) -> \( z = 1.00 \)
- Probability \( 0.98 \) -> \( z = 2.00 \)
Thus, the final answer is:
[tex]\[
[0.0, 0.0, 1.0, 2.0]
\][/tex]
