To find the percentage of oranges weighing between 9 oz and 10 oz, we need to calculate the area under the normal distribution curve between these two values. To do this, we will standardize the values and use the standard normal distribution (Z-distribution).

Given:

• Mean ($\mu$) = 8 oz
• Standard deviation ($\sigma$) = 1.5 oz
• We want to find the percentage of oranges between 9 oz and 10 oz.

Step 1: Standardize the values (convert to Z-scores)

The Z-score is calculated using the formula:

$Z = \frac{X - \mu}{\sigma}$

Where:

• $X$ is the value we are standardizing (9 oz and 10 oz),
• $\mu$ is the mean,
• $\sigma$ is the standard deviation.

For 9 oz: $Z_1 = \frac{9 - 8}{1.5} = \frac{1}{1.5} \approx 0.67$

For 10 oz: $Z_2 = \frac{10 - 8}{1.5} = \frac{2}{1.5} \approx 1.33$

Step 2: Find the area under the normal curve between these Z-scores

We will now look up the probabilities associated with the Z-scores from the standard normal distribution table (or use a calculator).

• The probability for $Z_1 = 0.67$ is approximately 0.7486.
• The probability for $Z_2 = 1.33$ is approximately 0.9082.

Step 3: Calculate the difference in probabilities

The percentage of oranges weighing between 9 oz and 10 oz is the difference in probabilities:

$P(9 \, \text{oz} \leq X \leq 10 \, \text{oz}) = P(Z_2) - P(Z_1) = 0.9082 - 0.7486 = 0.1596$

Step 4: Convert to percentage

To express this as a percentage:

$0.1596 \times 100 \approx 15.96\%$

Thus, about 16.0% of the oranges weigh between 9 oz and 10 oz.

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Would you like more details on how to calculate Z-scores or any further clarification? Here are some related questions:

1. How do you calculate probabilities using a Z-score table?
2. What is the concept of normal distribution in statistics?
3. How can you use a calculator to find areas under the normal curve?
4. What is the relationship between the mean and standard deviation in a normal distribution?
5. How can you apply normal distribution in real-world scenarios?

Tip: Always double-check the Z-scores for accuracy when using standard normal distribution tables or calculators to ensure correct probability values!