To find the probability that a data value in a normal distribution lies between the z-scores of -1.03 and -0.33, we can follow these steps:
1. Understand the Z-Scores: The z-scores are standardized scores that tell us how many standard deviations away a value is from the mean of the distribution. Here, we need the areas (probabilities) under the normal curve corresponding to the z-scores -1.03 and -0.33.
2. Find the Cumulative Probability for Each Z-Score:
- For a z-score of -1.03, we look up the cumulative probability, which gives us the probability that a value is less than or equal to -1.03. This cumulative probability is approximately 0.1515 or 15.15%.
- For a z-score of -0.33, we look up the cumulative probability, which gives us the probability that a value is less than or equal to -0.33. This cumulative probability is approximately 0.3707 or 37.07%.
3. Calculate the Probability Between the Two Z-Scores:
- We find the difference between the cumulative probabilities to determine the probability that a value lies between the two z-scores.
- Thus, we subtract the cumulative probability at z = -1.03 from the cumulative probability at z = -0.33:
[tex]\[
0.3707 - 0.1515 = 0.2192
\][/tex]
- Therefore, the probability that a value lies between the z-scores of -1.03 and -0.33 is approximately 0.2192 or 21.92%.
4. Convert to Percentage and Round:
- Convert this probability to a percentage by multiplying by 100:
[tex]\[
0.2192 \times 100 = 21.92\%
\][/tex]
- Round this result to the nearest tenth of a percent:
[tex]\[
21.9\%
\][/tex]
Given these calculations, the answer to the question is:
C. 21.9%
