Answer :
To solve this problem, let's create a table that correlates z-scores with their respective probabilities. The z-score measures how many standard deviations an element is from the mean of a distribution. The probability associated with each z-score indicates the likelihood that a value falls below that particular z-score in a standard normal distribution.
Here is the detailed step-by-step solution for the provided values:
1. Identify the z-scores and corresponding probabilities: We will create a table where each row shows a z-score and its corresponding cumulative probability from a standard normal distribution.
2. Create the table: Let's list each z-score and its probability. The z-score values range from 0.00 to 1.36, and for each z-score, there is an associated probability.
[tex]\[ \begin{array}{|c|c|} \hline \text{z} & \text{Probability} \\ \hline 0.00 & 0.5000 \\ \hline 0.25 & 0.5987 \\ \hline 0.35 & 0.6368 \\ \hline 0.45 & 0.6736 \\ \hline 1.00 & 0.8413 \\ \hline 1.26 & 0.8961 \\ \hline 1.35 & 0.9115 \\ \hline 1.36 & 0.9131 \\ \hline \end{array} \][/tex]
3. Interpretation: Each value in the "Probability" column represents the cumulative probability from the left of the standard normal distribution curve up to the corresponding z-score:
- For [tex]\( z = 0.00 \)[/tex], the probability is 0.5000.
- For [tex]\( z = 0.25 \)[/tex], the probability is 0.5987.
- For [tex]\( z = 0.35 \)[/tex], the probability is 0.6368.
- For [tex]\( z = 0.45 \)[/tex], the probability is 0.6736.
- For [tex]\( z = 1.00 \)[/tex], the probability is 0.8413.
- For [tex]\( z = 1.26 \)[/tex], the probability is 0.8961.
- For [tex]\( z = 1.35 \)[/tex], the probability is 0.9115.
- For [tex]\( z = 1.36 \)[/tex], the probability is 0.9131.
This table provides a clear reference for understanding the relationship between z-scores and their cumulative probabilities in a standard normal distribution. The higher the z-score, the higher the cumulative probability, indicating the proportion of data points falling below that z-score.
Here is the detailed step-by-step solution for the provided values:
1. Identify the z-scores and corresponding probabilities: We will create a table where each row shows a z-score and its corresponding cumulative probability from a standard normal distribution.
2. Create the table: Let's list each z-score and its probability. The z-score values range from 0.00 to 1.36, and for each z-score, there is an associated probability.
[tex]\[ \begin{array}{|c|c|} \hline \text{z} & \text{Probability} \\ \hline 0.00 & 0.5000 \\ \hline 0.25 & 0.5987 \\ \hline 0.35 & 0.6368 \\ \hline 0.45 & 0.6736 \\ \hline 1.00 & 0.8413 \\ \hline 1.26 & 0.8961 \\ \hline 1.35 & 0.9115 \\ \hline 1.36 & 0.9131 \\ \hline \end{array} \][/tex]
3. Interpretation: Each value in the "Probability" column represents the cumulative probability from the left of the standard normal distribution curve up to the corresponding z-score:
- For [tex]\( z = 0.00 \)[/tex], the probability is 0.5000.
- For [tex]\( z = 0.25 \)[/tex], the probability is 0.5987.
- For [tex]\( z = 0.35 \)[/tex], the probability is 0.6368.
- For [tex]\( z = 0.45 \)[/tex], the probability is 0.6736.
- For [tex]\( z = 1.00 \)[/tex], the probability is 0.8413.
- For [tex]\( z = 1.26 \)[/tex], the probability is 0.8961.
- For [tex]\( z = 1.35 \)[/tex], the probability is 0.9115.
- For [tex]\( z = 1.36 \)[/tex], the probability is 0.9131.
This table provides a clear reference for understanding the relationship between z-scores and their cumulative probabilities in a standard normal distribution. The higher the z-score, the higher the cumulative probability, indicating the proportion of data points falling below that z-score.
