The average weight of apples in a batch is 150 grams with a standard deviation of 10 grams. Assuming the weights follow a normal distribution, what percentage of apples weigh more than 170 grams?

Note:
1- The answer should be a number between 0 and 1. For instance, if the answer is 32% you should enter 0.32.
2- Enter the answer accurate to three decimal points.



To find the percentage of apples that weigh more than 170 grams, we can use the properties of a normal distribution and find the z-score corresponding to 170 grams.

### Solution:

1. **Calculate the z-score**:
   The z-score formula is:
   \[
   z = \frac{X - \mu}{\sigma}
   \]
   where:
   - \( X = 170 \) grams (the weight we are interested in),
   - \( \mu = 150 \) grams (the mean),
   - \( \sigma = 10 \) grams (the standard deviation).

   Plugging in the values:
   \[
   z = \frac{170 - 150}{10} = \frac{20}{10} = 2
   \]

2. **Find the probability**:
   The z-score of 2 corresponds to a cumulative probability of approximately 0.9772. This means that about 97.72% of the apples weigh **170 grams or less**.

3. **Calculate the percentage greater than 170 grams**:
   To find the percentage of apples weighing more than 170 grams:
   \[
   P(X > 170) = 1 - 0.9772 = 0.0228
   \]

### Answer:
The percentage of apples that weigh more than 170 grams is approximately **0.023 (or 2.3%)**.

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Would you like further clarification on any of the steps?

1. How do we find z-scores for values not exactly on standard tables?
2. How would changing the standard deviation affect the percentage greater than a given weight?
3. How is a cumulative distribution different from a probability density?
4. What is the significance of one standard deviation in normal distributions?
5. How do we interpret probabilities in a real-world context?

**Tip:** Remember, in normal distributions, around 68% of data lies within one standard deviation of the mean.

The average weight of apples in a batch is 150 grams with a standard deviation of 10 grams. Assuming the weights follow a normal distribution, what percentage of apples weigh more than 170 grams?

Learn how to calculate the percentage of apples in a batch that weigh more than 170 grams, given an average weight of 150 grams and a standard deviation of 10 grams. This problem covers probability, Z-scores, and cumulative probability concepts, ideal for Grades 10-12.

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