To determine the probability that a randomly selected student has a score between 350 and 550 on a standardized test with a mean of 500 and a standard deviation of 110, follow these steps:
1. Identify the Given Information:
- Mean ([tex]\(\mu\)[/tex]) of the test scores = 500
- Standard Deviation ([tex]\(\sigma\)[/tex]) of the test scores = 110
- Lower score boundary = 350
- Upper score boundary = 550
2. Calculate the Z-Scores:
- The Z-score for a given value [tex]\(X\)[/tex] is calculated using the formula:
[tex]\[
Z = \frac{X - \mu}{\sigma}
\][/tex]
- For the lower score (350):
[tex]\[
Z_{lower} = \frac{350 - 500}{110} \approx -1.36
\][/tex]
- For the upper score (550):
[tex]\[
Z_{upper} = \frac{550 - 500}{110} \approx 0.45
\][/tex]
3. Find the Corresponding Probabilities:
- Using the Z-table, we find:
- The probability corresponding to [tex]\(Z = -1.36\)[/tex] is approximately 0.159.
- The probability corresponding to [tex]\(Z = 0.45\)[/tex] is approximately 0.8413.
4. Calculate the Probability of a Score Between the Given Bounds:
- To find the probability of a score being between the Z-scores, subtract the lower Z-score probability from the upper Z-score probability:
[tex]\[
\text{Probability} = P(Z_{upper}) - P(Z_{lower}) = 0.8413 - 0.159 = 0.6823
\][/tex]
Therefore, the probability that a randomly selected student has a score between 350 and 550 is approximately 0.6823 or 68.23%.
Among the provided options (9%, 24%, 59%, 91%), none of them directly matches our calculated probability of 68.23%. This might indicate that the correct answer isn’t explicitly listed in the given options.
