The length of pregnancy of a cow from conception to birth approximates a normal
distibution with a mean of 283 days and standard deviation of 12 days.

What is the probability of pregnancy of a cow will last within 285 days

To solve this problem, we need to determine the probability that a cow's pregnancy will last within 285 days, given that the pregnancy duration follows a normal distribution with a mean (\(\mu\)) of 283 days and a standard deviation (\(\sigma\)) of 12 days.

### Step 1: Calculate the Z-score
The Z-score is a measure of how many standard deviations an element is from the mean. The Z-score is calculated using the formula:

\[
Z = \frac{X - \mu}{\sigma}
\]

Where:
- \(X\) is the value we are interested in (285 days),
- \(\mu\) is the mean (283 days),
- \(\sigma\) is the standard deviation (12 days).

Substituting the values:

\[
Z = \frac{285 - 283}{12} = \frac{2}{12} \approx 0.167
\]

### Step 2: Determine the Probability
Next, we need to find the probability corresponding to this Z-score. We can use the standard normal distribution table or a calculator to find this.

The Z-score of 0.167 corresponds to a cumulative probability of approximately 0.5665.

### Interpretation
This means that there is a 56.65% probability that a cow's pregnancy will last less than or equal to 285 days.

Since you are asking about the probability of pregnancy lasting "within" 285 days (assuming you mean up to 285 days), the probability is 56.65%.

Would you like any more details or explanations?

Here are 5 related questions you might be interested in:
1. What is the probability that a pregnancy will last more than 290 days?
2. How can we calculate the probability of a pregnancy lasting between 270 and 290 days?
3. What is the probability of a pregnancy lasting exactly 283 days?
4. How does changing the standard deviation affect the probability calculations?
5. Can you find the range of days where 95% of the pregnancies fall?

**Tip:** When dealing with probabilities in normal distributions, the Z-score is key to finding the cumulative probabilities, which can then be looked up in standard normal distribution tables.

What is the probability of pregnancy of a cow will last within 285 days?

Discover how to calculate the probability that a cow's pregnancy will last within 285 days using normal distribution. This problem involves key statistical concepts such as the Z-score formula and is suitable for students in Grades 11-12.

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