We are dealing with a normally distributed set of test scores with a given mean and standard deviation. Follow the steps below to determine the probability that a randomly selected student scores between 350 and 550.
1. Identify the given parameters:
- Mean ([tex]\(\mu\)[/tex]): 500
- Standard Deviation ([tex]\(\sigma\)[/tex]): 110
- Lower bound: 350
- Upper bound: 550
2. Convert the raw scores to z-scores:
The z-score for a value [tex]\(X\)[/tex] is given by:
[tex]\[
z = \frac{X - \mu}{\sigma}
\][/tex]
For the lower bound [tex]\(X = 350\)[/tex]:
[tex]\[
z_{\text{lower}} = \frac{350 - 500}{110} = \frac{-150}{110} = -1.3636
\][/tex]
For the upper bound [tex]\(X = 550\)[/tex]:
[tex]\[
z_{\text{upper}} = \frac{550 - 500}{110} = \frac{50}{110} = 0.4545
\][/tex]
3. Use the z-score table to find the corresponding probabilities:
- For [tex]\(z = -1.36\)[/tex], the table value is approximately 0.0869. This means that the probability of a value being less than 350 is approximately [tex]\(0.0869\)[/tex].
- For [tex]\(z = 0.45\)[/tex], the table value is approximately 0.6736. This means that the probability of a value being less than 550 is approximately [tex]\(0.6736\)[/tex].
4. Calculate the probability that the score is between 350 and 550:
The probability of a student scoring between two z-scores is the difference between their cumulative probabilities.
[tex]\[
\text{Probability} = P(X < 550) - P(X < 350) = 0.6736 - 0.5184 = 0.1552
\][/tex]
Thus, the probability that a randomly selected student scores between 350 and 550 is approximately [tex]\(0.1552\)[/tex] or [tex]\(15.52\%\)[/tex].
