To find the probability that a randomly selected value from a normally distributed set of data falls between 227.3 and 247.4, given a mean ([tex]\(\mu\)[/tex]) of 172.1 and a standard deviation ([tex]\(\sigma\)[/tex]) of 25.1, follow these steps:
1. Standardization: Convert the values 227.3 and 247.4 into their corresponding z-scores.
The z-score is calculated using the formula:
[tex]\[
z = \frac{X - \mu}{\sigma}
\][/tex]
where [tex]\( X \)[/tex] is the value from the dataset, [tex]\( \mu \)[/tex] is the mean, and [tex]\( \sigma \)[/tex] is the standard deviation.
2. Calculation of z-scores:
- For 227.3:
[tex]\[
z_{\text{lower}} = \frac{227.3 - 172.1}{25.1} = \frac{55.2}{25.1} \approx 2.1992
\][/tex]
- For 247.4:
[tex]\[
z_{\text{upper}} = \frac{247.4 - 172.1}{25.1} = \frac{75.3}{25.1} \approx 3.0
\][/tex]
3. Finding the probabilities: Use the cumulative distribution function (CDF) of the standard normal distribution to find the probabilities corresponding to these z-scores.
- The probability [tex]\( P(z < z_{\text{upper}}) \)[/tex] corresponds to the area under the standard normal curve to the left of [tex]\( z = 3.0 \)[/tex].
- The probability [tex]\( P(z < z_{\text{lower}}) \)[/tex] corresponds to the area under the standard normal curve to the left of [tex]\( z = 2.1992 \)[/tex].
4. Subtract the probabilities: The probability that [tex]\( X \)[/tex] falls between 227.3 and 247.4 is the difference between these two probabilities.
[tex]\[
P(227.3 < X < 247.4) = P(z < z_{\text{upper}}) - P(z < z_{\text{lower}})
\][/tex]
5. Result:
[tex]\[
P(227.3 < X < 247.4) \approx 0.0126
\][/tex]
Therefore, the probability that a randomly selected value from this normal distribution is between 227.3 and 247.4 is [tex]\( \boxed{0.0126} \)[/tex].
