The acceptable level for insect filth in a certain food item is 5 insect fragments (larvae, eggs, body parts, and so on) per 10 grams. A simple random sample of 40 ten-gram portions of the food item is obtained and results in a sample mean of x overbar = 5.5
insect fragments per ten-gram portion.

What is the probability a simple random sample of 40 ten-gram portions of the food item results in a mean of at least 5.5 insect fragments?


To solve this problem, we can apply the concept of the **Central Limit Theorem (CLT)**, which is useful when dealing with sample means from a population. Given the data, we need to calculate the probability that the sample mean is **at least 5.5 insect fragments per ten-gram portion**.

### Step 1: Identify the Given Information
We are given:
- The population mean (\(\mu\)) is **5 insect fragments per ten-gram portion**.
- The sample size (\(n\)) is **40**.
- The sample mean (\(\bar{x}\)) is **5.5 insect fragments per ten-gram portion**.
- We need to find the probability that the sample mean is **at least 5.5**.

### Step 2: Population Standard Deviation
We are not explicitly given the standard deviation (\(\sigma\)) of the population. In such cases, we typically assume a value based on the context or further information. For now, I will assume that \(\sigma\) is known or provided. Let me know if you want to use a specific value or need further clarification on this step.

### Step 3: Use the Central Limit Theorem
The Central Limit Theorem states that for large samples (typically \(n > 30\)), the distribution of the sample mean \(\bar{x}\) is approximately normal with:
- Mean: \(\mu_{\bar{x}} = \mu = 5\)
- Standard deviation (Standard error): \(\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}\)

### Step 4: Calculate the Z-Score
The Z-score formula for the sample mean is:

\[
Z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}
\]

### Step 5: Find the Probability
Once we calculate the Z-score, we can use the standard normal distribution table (or a calculator) to find the probability associated with that Z-score, which will give us the probability that the sample mean is **at least 5.5** insect fragments.

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Could you provide the standard deviation \(\sigma\) or let me know if I should proceed with an assumption?

The acceptable level for insect filth in a certain food item is 5 insect fragments (larvae, eggs, body parts, and so on) per 10 grams. A simple random sample of 40 ten-gram portions of the food item is obtained and results in a sample mean of x̄ = 5.5 insect fragments per ten-gram portion. What is the probability a simple random sample of 40 ten-gram portions of the food item results in a mean of at least 5.5 insect fragments?

Calculate the probability that a sample mean of 40 ten-gram portions of food exceeds 5.5 insect fragments per portion using the Central Limit Theorem. This problem covers key statistical concepts such as Z-scores, probability distribution, and the Central Limit Theorem.

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