What is the probability that the average bacteria count for the 42 samples is between 2495 and 2665 bacteria per milliliter?

Steps to Solve

1. Understand the Normal Distribution Parameters

You have the mean ($\mu$) and standard deviation ($\sigma$) for the normal distribution. Here, $\mu = 2700$ and $\sigma = 61.72$, given in the problem statement.

2. Standardize the Values for the Probability Calculation

To find the probability that the average bacteria count falls between 2495 and 2665, we need to convert these values into z-scores using the formula:

$$ z = \frac{x - \mu}{\sigma / \sqrt{n}} $$

where $n$ is the sample size (42 in this case).

3. Calculate z-scores for the Boundaries

Calculate the z-score for 2495:

$$ z_1 = \frac{2495 - 2700}{\frac{61.72}{\sqrt{42}}} $$

Calculate the z-score for 2665:

$$ z_2 = \frac{2665 - 2700}{\frac{61.72}{\sqrt{42}}} $$

4. Find the z-score Values

Use a calculator or statistical table to compute:

• For $z_1$
• For $z_2$

5. Calculate the Probability Using z-scores

Find the probabilities corresponding to the calculated z-scores using the standard normal distribution:

$$ P(Z < z_1) \text{ and } P(Z < z_2) $$

Then, the desired probability is:

$$ P(z_1 < Z < z_2) = P(Z < z_2) - P(Z < z_1) $$

6. Final Calculation

Substitute the values found previously and calculate the final probability.