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A distribution of values is normal with a mean of 172.1 and a standard deviation of 25.1.

Find the probability that a randomly selected value is between 227.3 and 247.4.

[tex] P(227.3 \ \textless \ X \ \textless \ 247.4) = [/tex]

[tex] \square [/tex]

Enter your answer as a number accurate to 4 decimal places.


Sagot :

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].