Suppose the weight of mandarins in a large crate has a distribution with mean 101 grams and standard deviation 15.4 grams. Consider the process of picking 5 mandarins at random from the crate and putting them in a bag. Let X be the mean weight of the mandarins in the bag. Assuming the weights of mandarins are Normally distributed, the probability that the mean weight of the mandarins in the bag is at least 118 grams is

To solve this problem, we need to calculate the probability that the mean weight \( X \) of the 5 mandarins in the bag is at least 118 grams, given that the weights of the mandarins are Normally distributed with a mean of 101 grams and a standard deviation of 15.4 grams.

### Step 1: Identify the distribution of the sample mean \( X \)
Since the individual weights of mandarins are Normally distributed, the sample mean \( X \) is also Normally distributed. The mean of \( X \) is the same as the population mean, and the standard deviation of \( X \) (also called the standard error) is given by:

\[
\text{Standard error of } X = \frac{\sigma}{\sqrt{n}}
\]

where:
- \( \sigma = 15.4 \) grams (population standard deviation)
- \( n = 5 \) (number of mandarins in the sample)

\[
\text{Standard error of } X = \frac{15.4}{\sqrt{5}} \approx 6.88 \text{ grams}
\]

### Step 2: Standardize the problem
We need to find the probability that \( X \geq 118 \) grams. To do this, we first standardize \( X \) using the Z-score formula:

\[
Z = \frac{X - \mu}{\text{Standard error of } X}
\]

Substituting the values:

\[
Z = \frac{118 - 101}{6.88} \approx \frac{17}{6.88} \approx 2.47
\]

### Step 3: Find the probability
The probability we seek is \( P(X \geq 118) \), which is equivalent to \( P(Z \geq 2.47) \).

Using the standard normal distribution table, we find the probability corresponding to \( Z = 2.47 \). The table gives the probability \( P(Z \leq 2.47) \), which is about 0.9932. Therefore:

\[
P(Z \geq 2.47) = 1 - P(Z \leq 2.47) = 1 - 0.9932 = 0.0068
\]

### Final Answer:
The probability that the mean weight of the mandarins in the bag is at least 118 grams is approximately **0.0068**.

Would you like to explore any specific details or have further questions?

Here are some related questions to consider:
1. How would the probability change if we picked 10 mandarins instead of 5?
2. What is the probability that the mean weight is less than 95 grams?
3. How would the calculation change if the weights were not Normally distributed?
4. What would the standard error be if we picked 20 mandarins?
5. Can you calculate the probability for the mean weight to be between 100 and 110 grams?

**Tip:** Remember, when working with sample means, the standard deviation of the sample mean (the standard error) decreases as the sample size increases.

Explore the probability that the mean weight of 5 mandarins randomly picked from a crate, with a distribution mean of 101 grams and standard deviation of 15.4 grams, is at least 118 grams. Learn about Normal distribution, standard error calculations, and Z-score formula applications.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation