Math Problem Statement

Suppose the weight of bananas in a large crate has a distribution with mean 158 grams and standard deviation 19.2 grams. Consider the process of picking 8 bananas at random from the crate and putting them in a bag. Let X be the mean weight of the bananas in the bag. Assuming the weights of bananas are Normally distributed, the probability that the mean weight of the bananas in the bag is at least 174 grams is

Solution

To solve this problem, we'll use the concept of the sampling distribution of the sample mean XX. Given that the weights of the bananas are Normally distributed, the sample mean XX will also be Normally distributed. Here’s a step-by-step breakdown:

1. Parameters of the Population

  • Mean weight of bananas μ=158\mu = 158 grams
  • Standard deviation of the weights σ=19.2\sigma = 19.2 grams

2. Parameters of the Sampling Distribution

For a sample size n=8n = 8, the mean of the sampling distribution of XX (the sample mean) is the same as the population mean: μX=μ=158 grams\mu_X = \mu = 158 \text{ grams} The standard deviation of the sampling distribution of XX (standard error) is: σX=σn=19.286.787 grams\sigma_X = \frac{\sigma}{\sqrt{n}} = \frac{19.2}{\sqrt{8}} \approx 6.787 \text{ grams}

3. Standardizing the Problem

We want to find the probability that the mean weight XX of the bananas in the bag is at least 174 grams. This requires finding: P(X174)P(X \geq 174) First, we standardize this using the Z-score formula: Z=XμXσX=1741586.7872.36Z = \frac{X - \mu_X}{\sigma_X} = \frac{174 - 158}{6.787} \approx 2.36

4. Finding the Probability

Now, we need to find the probability corresponding to Z2.36Z \geq 2.36 using the standard normal distribution.

Using standard normal distribution tables or a calculator: P(Z2.36)1P(Z2.36)10.9909=0.0091P(Z \geq 2.36) \approx 1 - P(Z \leq 2.36) \approx 1 - 0.9909 = 0.0091

5. Conclusion

The probability that the mean weight of the bananas in the bag is at least 174 grams is approximately 0.0091, or 0.91%.

Would you like more details or have any questions?

Here are 5 related questions you might find interesting:

  1. What is the probability that the mean weight of 8 bananas is less than 150 grams?
  2. How would the probability change if the sample size were increased to 16 bananas?
  3. What is the probability that the mean weight of 8 bananas is exactly 158 grams?
  4. How would the calculation change if the distribution of weights was not normal?
  5. Can you find the 95% confidence interval for the mean weight of 8 bananas?

Tip: Remember that the standard deviation of the sample mean decreases as the sample size increases, making the distribution of the sample mean narrower.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Normal Distribution
Sampling Distribution
Z-score

Formulas

Z-score formula

Theorems

-

Suitable Grade Level

Advanced High School or College