Suppose the weight of tomatoes in a large crate has a distribution with mean 141 grams and standard deviation 25.3 grams. Consider the process of picking 6 tomatoes at random from the crate and putting them in a bag. Let X be the mean weight of the tomatoes in the bag. Assuming the weights of tomatoes are Normally distributed, the probability that the mean weight of the tomatoes in the bag is at least 166 grams is

To solve this problem, we need to determine the probability that the mean weight \( X \) of the tomatoes in the bag is at least 166 grams. Here’s the step-by-step process:

### Step 1: Identify the parameters
- Mean of the population \( \mu = 141 \) grams
- Standard deviation of the population \( \sigma = 25.3 \) grams
- Sample size \( n = 6 \)

### Step 2: Find the standard error of the mean
The standard error of the mean \( \text{SE} \) is calculated as:
\[
\text{SE} = \frac{\sigma}{\sqrt{n}} = \frac{25.3}{\sqrt{6}} \approx 10.326
\]

### Step 3: Standardize the desired value
We want to find the probability that the sample mean \( X \) is at least 166 grams. First, we convert 166 grams to the corresponding z-score:
\[
z = \frac{X - \mu}{\text{SE}} = \frac{166 - 141}{10.326} \approx \frac{25}{10.326} \approx 2.422
\]

### Step 4: Find the probability
Now, we need to find the probability that \( X \geq 166 \). This corresponds to finding \( P(Z \geq 2.422) \), where \( Z \) is a standard normal variable.

Using standard normal distribution tables or a calculator, we find:
\[
P(Z \geq 2.422) \approx 0.0077
\]

### Final Answer
The probability that the mean weight of the tomatoes in the bag is at least 166 grams is approximately **0.0077**.

Would you like more details on any part of the solution? 

Here are some related questions you might find helpful:
1. How does increasing the sample size affect the standard error?
2. What is the probability that the mean weight of the tomatoes in the bag is less than 130 grams?
3. How would the probability change if the standard deviation was higher?
4. Can we use this method if the weights of tomatoes are not Normally distributed?
5. What is the probability that a randomly selected tomato weighs more than 166 grams?

**Tip:** When dealing with normally distributed variables, always check if the sample size is large enough for the Central Limit Theorem to apply if the population distribution is not normal.

Explore how to calculate the probability that the mean weight of tomatoes in a bag is at least 166 grams using normal distribution and standard error concepts. This problem involves a crate of tomatoes with specified mean and standard deviation, and picking a sample of tomatoes to determine the mean weight in a bag.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation