Answer :
To solve the problem of determining the probabilities related to the weights of cement bags, we'll make use of the properties of the normal distribution. The weights are normally distributed with a mean ([tex]\mu[/tex]) of 39.3 kg and a standard deviation ([tex]\sigma[/tex]) of 0.9 kg. This problem involves finding probabilities using the standard normal distribution (also known as the Z-distribution).
(a) Find the probability that the bag weighs more than 40 kg.
To find this probability, we first need to convert the weight of 40 kg to a z-score using the formula:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
Where:
- [tex]X[/tex] is the value we are interested in (40 kg in this case),
- [tex]\mu[/tex] is the mean (39.3 kg),
- [tex]\sigma[/tex] is the standard deviation (0.9 kg).
Substituting these values into the formula gives:
[tex]Z = \frac{40 - 39.3}{0.9} = \frac{0.7}{0.9} \approx 0.78[/tex]
Next, we use the standard normal distribution table (or a calculator with statistical functions) to find the probability that a Z-score is greater than 0.78. This value represents the area to the right of 0.78 under the normal curve. Looking this up (or calculating it), we get:
[tex]P(Z > 0.78) \approx 1 - 0.7823 = 0.2177[/tex]
Thus, the probability that a randomly selected bag weighs more than 40 kg is approximately 0.2177 or 21.77%.
(b) Find the probability that the bag weighs between 38 kg and 41 kg.
For this part, we need to find the probability that the bag's weight lies between 38 kg and 41 kg. This is the difference between the probability of the bag weighing less than 41 kg and the probability of it weighing less than 38 kg.
First, we convert these weights to z-scores:
For 41 kg:
[tex]Z = \frac{41 - 39.3}{0.9} = \frac{1.7}{0.9} \approx 1.89[/tex]
For 38 kg:
[tex]Z = \frac{38 - 39.3}{0.9} = \frac{-1.3}{0.9} \approx -1.44[/tex]
Using the standard normal distribution table (or a calculator), we find:
- The probability that Z < 1.89 is approximately 0.9706.
- The probability that Z < -1.44 is approximately 0.0749.
Thus, the probability that the bag weighs between 38 kg and 41 kg is:
[tex]P(38 < X < 41) = P(Z < 1.89) - P(Z < -1.44)[/tex]
[tex]P(38 < X < 41) \approx 0.9706 - 0.0749 = 0.8957[/tex]
Therefore, the probability that a randomly selected bag weighs between 38 kg and 41 kg is approximately 0.8957 or 89.57%.