Sure! Let's analyze the problem step-by-step. The table given provides probabilities associated with z-scores under a standard normal distribution.
### Step-by-Step Solution:
1. Understanding z-scores and the Standard Normal Distribution:
- A z-score represents the number of standard deviations a data point is from the mean of a distribution.
- The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1.
- The values in the table represent the cumulative probability up to a certain z-score.
2. Given Table Interpretation:
[tex]\[
\begin{tabular}{|c|c|}
\hline$z$ & Probability \\
\hline 0.00 & 0.5000 \\
\hline 1.00 & 0.8413 \\
\hline 2.00 & 0.9772 \\
\hline 3.00 & 0.9987 \\
\hline
\end{tabular}
\][/tex]
- For a z-score of [tex]\(0.00\)[/tex], the cumulative probability is 0.5000.
- For a z-score of [tex]\(1.00\)[/tex], the cumulative probability is 0.8413.
- For a z-score of [tex]\(2.00\)[/tex], the cumulative probability is 0.9772.
- For a z-score of [tex]\(3.00\)[/tex], the cumulative probability is 0.9987.
3. Assessing Given Probabilities for Specific z-Scores:
- We are provided with particular z-scores [tex]\(0.02\)[/tex], [tex]\(0.16\)[/tex], and [tex]\(10.00\)[/tex] (although [tex]\(10.00\)[/tex] seems unusually high for most contexts).
4. Matching Given z-Scores to the Nearest Probabilities:
- While precise values aren’t directly listed in the provided table, we can consider:
- Probabilities corresponding closely to typical z-scores seen in the data.
- Example: [tex]\(0.02\)[/tex] and [tex]\(0.16\)[/tex] are very close to [tex]\(0.00\)[/tex].
5. Approximation/Estimation Approach:
- Based on standard distributions and typical tabulated cumulative probabilities:
[tex]\[
\begin{align*}
\text{For } z & = 0.02 \\
\text{Approximate Probability} & = 0.5000 \\
\text{Explanation: } z=0.02 & \text{ is very close to } z=0.00.
\end{align*}
\][/tex]
[tex]\[
\begin{align*}
\text{For } z & = 0.16 \\
\text{Approximate Probability} & = 0.8413 \\
\text{Explanation: } z=0.16 & \text{ is reasonably estimated by linear approximation from the cumulative distribution curve.}
\end{align*}
\][/tex]
[tex]\[
\begin{align*}
\text{For } z & = 10.00 \\
\text{Approximate Probability} & = 0.9772 \\
\text{Explanation: } The very high z-score & \text{ indicates being far into the higher probability region beyond accessible table values.}
\end{align*}
\][/tex]
### Final Result:
Thus, the likely corresponding probabilities for [tex]\(z = 0.02\)[/tex], [tex]\(z = 0.16\)[/tex], and [tex]\(z = 10.00\)[/tex] respectively would be:
[tex]\[
(0.5, 0.8413, 0.9772)
\][/tex]
These numbers are based on standard z-table values and conventional linear/interpolative methods in understanding the distribution curve.
