The scores of the students on a standardized test are normally distributed, with a mean of 500 and a standard deviation of 110. What is the probability that a randomly selected student has a score between 350 and 550? Use the portion of the standard normal table below to help answer the question.

\begin{tabular}{|c|c|}
\hline
[tex]$z$[/tex] & 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{tabular}

A. 9\%

B. 24\%

C. 59\%



Answer :

To determine the probability that a randomly selected student scores between 350 and 550 on a standardized test, we need to follow these steps:

1. Understand the distribution parameters:
- Mean ([tex]\( \mu \)[/tex]) = 500
- Standard deviation ([tex]\( \sigma \)[/tex]) = 110

2. Convert the raw scores to z-scores:
The z-score formula is:
[tex]\[ z = \frac{X - \mu}{\sigma} \][/tex]
where [tex]\( X \)[/tex] is the raw score.

3. Calculate the z-score for the lower bound (350):
[tex]\[ z_{\text{lower}} = \frac{350 - 500}{110} = \frac{-150}{110} \approx -1.36 \][/tex]

4. Calculate the z-score for the upper bound (550):
[tex]\[ z_{\text{upper}} = \frac{550 - 500}{110} = \frac{50}{110} \approx 0.45 \][/tex]

5. Use the standard normal table to find the probabilities:
- From the table, the probability corresponding to [tex]\( z = -1.36 \)[/tex] is approximately 0.0869 (since it is the cumulative probability up to [tex]\( z = -1.36 \)[/tex]). Note that we actually need:
[tex]\[ P(Z < z_{\text{lower}}) = 0.1590 \quad (\text{since the table gives us the probability for the positive counterpart and we subtract from 1}) \][/tex]
- The probability corresponding to [tex]\( z = 0.45 \)[/tex] is approximately 0.6736.

6. Calculate the probability that the score is between the two z-scores:
[tex]\[ \text{Probability} = P(z_{\text{upper}}) - P(z_{\text{lower}}) = 0.6736 - 0.1590 = 0.5146 \approx 0.6823 \][/tex]

Thus, the probability that a randomly selected student scores between 350 and 550 on the test is approximately [tex]\( 68.23\% \)[/tex].