Suppose the mass of students, in kg is N(68; 9) distributed. Find the probability of students with the mass between 64 and 67 kg.
To find the probability of students with a mass between 64 and 67 kg, we need to calculate the cumulative probability at the upper bound of 67 kg and subtract the cumulative probability at the lower bound of 64 kg.
First, we need to convert the mass values to z-scores using the formula:
\[ Z = \frac{X - \mu}{\sigma} \]
Where:
- X = mass value
- μ = mean mass (68 kg)
- σ = standard deviation (9 kg)
For X = 67 kg:
\[ Z_{67} = \frac{67 - 68}{9} = -0.111 \]
For X = 64 kg:
\[ Z_{64} = \frac{64 - 68}{9} = -0.444 \]
Next, we look up these z-scores in a standard normal distribution table (or use a calculator) to find the corresponding probabilities:
- Probability for Z = -0.111 is 0.4554
- Probability for Z = -0.444 is 0.3289
Now, we calculate the probability of students with mass between 64 and 67 kg:
\[ P(64 < X < 67) = P(X < 67) - P(X < 64) \]
\[ = 0.4554 - 0.3289 \]
\[ = 0.1265 \]
Therefore, the probability of students with mass between 64 and 67 kg is approximately 0.1265 or 12.65%.