i just devised a new biscuit with a mean baking time of 7.1 minutes and a standard deviation of 2.5 minutes. of baking a sheet of 36 biscuits find the probability that the mean taking time for the sheet of 36 biscuits is no longer than 8.6 minutes

To find the probability that the mean baking time for a sheet of 36 biscuits is no longer than 8.6 minutes, we can approach this using the Central Limit Theorem (CLT). The CLT tells us that for a sufficiently large sample size, the sampling distribution of the sample mean approaches a normal distribution, even if the underlying data is not normally distributed.

Here’s the step-by-step process:

### Step 1: Set up the problem
We are given:
- Population mean (μ) = 7.1 minutes
- Population standard deviation (σ) = 2.5 minutes
- Sample size (n) = 36
- We want to find the probability that the mean of the sample (sheet of 36 biscuits) is no longer than 8.6 minutes.

### Step 2: Compute the standard error of the mean
The standard error (SE) is the standard deviation of the sample mean distribution and is calculated by:

\[
SE = \frac{\sigma}{\sqrt{n}} = \frac{2.5}{\sqrt{36}} = \frac{2.5}{6} = 0.4167
\]

### Step 3: Standardize the problem (find the Z-score)
We convert the sample mean of 8.6 minutes to a Z-score, which tells us how many standard errors this value is away from the population mean:

\[
Z = \frac{X - \mu}{SE} = \frac{8.6 - 7.1}{0.4167} = \frac{1.5}{0.4167} \approx 3.60
\]

### Step 4: Find the probability corresponding to the Z-score
Using a Z-table or standard normal distribution calculator, we look up the probability corresponding to a Z-score of 3.60.

The probability that \( Z \leq 3.60 \) is approximately 0.9998.

### Step 5: Interpret the result
This means that the probability that the mean baking time for a sheet of 36 biscuits is no longer than 8.6 minutes is approximately **99.98%**.

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Would you like more details or have any questions about this solution?

Here are 5 related questions you might want to explore:
1. What if the sample size were smaller, say 16 biscuits? How would that change the probability?
2. How does the Central Limit Theorem apply to different population distributions?
3. What is the probability that the baking time is longer than 8.6 minutes?
4. How does increasing the standard deviation affect the baking process?
5. How would you find the probability for a different mean, say 7.5 minutes?

**Tip**: The larger the sample size, the more the sample mean will tend to be close to the population mean, due to the Central Limit Theorem!

Discover how to calculate the probability that the mean baking time for a sheet of 36 biscuits is no longer than 8.6 minutes. Learn key statistical concepts such as the Central Limit Theorem, standard error, and Z-scores. This problem is ideal for students in Grades 10-12.

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