Westonci.ca is the premier destination for reliable answers to your questions, provided by a community of experts. Connect with a community of experts ready to provide precise solutions to your questions on our user-friendly Q&A platform. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.
Sagot :
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.
We hope this information was helpful. Feel free to return anytime for more answers to your questions and concerns. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Get the answers you need at Westonci.ca. Stay informed with our latest expert advice.