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The problem provided is about finding probabilities for certain weight ranges using a normal distribution. The information given includes:

- Mean (\(\mu\)) = 61 kg
- Standard deviation (\(\sigma\)) = 14 kg

We are asked to find the probability for three cases:
- (a) Probability that a woman's weight is between 50 and 75 kg.
- (b) Probability that a woman's weight is more than 45 kg.
- (c) Probability that a woman's weight is less than 50 kg.

These probabilities can be calculated using the Z-score formula for the normal distribution:
\[
Z = \frac{X - \mu}{\sigma}
\]
where \(X\) is the value of interest, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.

Let's calculate each probability step by step.

### (a) Probability of weight between 50 and 75 kg:
We need to find \(P(50 \leq X \leq 75)\).

1. Calculate the Z-scores for 50 and 75:
\[
Z_{50} = \frac{50 - 61}{14} = \frac{-11}{14} \approx -0.79
\]
\[
Z_{75} = \frac{75 - 61}{14} = \frac{14}{14} = 1.00
\]

2. Using the Z-table (or calculator), find the cumulative probabilities for these Z-scores:
- \(P(Z_{50}) \approx 0.2148\)
- \(P(Z_{75}) = 0.8413\)

Thus, the probability is:
\[
P(50 \leq X \leq 75) = P(Z_{75}) - P(Z_{50}) = 0.8413 - 0.2148 = 0.6265
\]

### (b) Probability of weight greater than 45 kg:
We need to find \(P(X > 45)\).

1. Calculate the Z-score for 45:
\[
Z_{45} = \frac{45 - 61}{14} = \frac{-16}{14} \approx -1.14
\]

2. Find the cumulative probability for \(Z_{45}\):
- \(P(Z_{45}) \approx 0.1271\)

Thus, the probability is:
\[
P(X > 45) = 1 - P(Z_{45}) = 1 - 0.1271 = 0.8729
\]

### (c) Probability of weight less than 50 kg:
We need to find \(P(X < 50)\).

1. We already calculated the Z-score for 50: \(Z_{50} \approx -0.79\).

2. The cumulative probability is:
- \(P(Z_{50}) \approx 0.2148\)

Thus, the probability is:
\[
P(X < 50) = 0.2148
\]

### Summary of Probabilities:
- (a) \(P(50 \leq X \leq 75) = 0.6265\)
- (b) \(P(X > 45) = 0.8729\)
- (c) \(P(X < 50) = 0.2148\)

Would you like further details on any of these steps or concepts?

Here are 5 related questions:
1. How do you interpret the Z-score in a normal distribution?
2. What is the role of the standard deviation in determining the shape of the distribution?
3. How would the probabilities change if the standard deviation was smaller?
4. What is the probability of a woman weighing exactly 61 kg in this scenario?
5. How can you find probabilities for non-standard normal distributions?

**Tip:** The total area under a normal distribution curve always equals 1, which corresponds to a 100% probability for all possible outcomes.

Rata-rata berat badan wanita berusia 18-24 tahun adalah 61 kg dengan standar deviasi 14 kg. Hitunglah peluang wanita memiliki berat:

a. 50-75 kg
b. >45 kg
c. <50 kg

Find the probability that a woman's weight (ages 18-24, mean 61 kg, standard deviation 14 kg) falls between 50-75 kg, is greater than 45 kg, or is less than 50 kg. Learn to apply the Z-score formula in a normal distribution to solve this problem.

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