Answer :
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.
### 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.
