To find the probability that a single randomly selected value from a normally distributed population is greater than 15.4, we need to standardize the value using the Z-score formula and then use the standard normal distribution to find the corresponding probability.

Step-by-Step Solution:

1. Calculate the Z-score for $X = 15.4$:

$Z = \frac{X - \mu}{\sigma}$

where:

• $\mu = 26.2$ (population mean)
• $\sigma = 85.3$ (population standard deviation)
• $X = 15.4$ (value of interest)

$Z = \frac{15.4 - 26.2}{85.3}$

$Z = \frac{-10.8}{85.3}$

$Z \approx -0.1265$

2. Find the probability corresponding to $Z \leq -0.1265$ using the standard normal distribution table or a calculator.

Using a Z-table or a standard normal distribution calculator, we find the cumulative probability for $Z \leq -0.1265$:

$P(Z \leq -0.1265) \approx 0.450$

3. Find the probability $P(X > 15.4)$:

Since the total area under the standard normal curve is 1, the probability of $Z > -0.1265$ is:

$P(Z > -0.1265) = 1 - P(Z \leq -0.1265)$

$P(Z > -0.1265) = 1 - 0.450$

$P(Z > -0.1265) \approx 0.5500$

Thus, the probability that a single randomly selected value is greater than 15.4 is approximately 0.5500.

Would you like more details or have any questions about this solution?

Here are 5 numbered questions for you to ask:

1. How do I interpret the Z-score in a standard normal distribution?
2. Can you explain how to use the Z-table to find probabilities?
3. What steps should I take if the population distribution is not normal?
4. How do I find probabilities for other ranges of values?
5. Can you show an example of calculating the Z-score for a different value?

Tip: Always double-check your Z-score calculations and ensure you're using the correct cumulative probability from the Z-table or standard normal distribution calculator.