Olive weights are classified according to a unique set of adjectives implying great size. For example, the mean weight of olives classified as 'Colossal' is 7.7 grams. Suppose a pa... Olive weights are classified according to a unique set of adjectives implying great size. For example, the mean weight of olives classified as 'Colossal' is 7.7 grams. Suppose a particular company’s crop of 'Colossal' olives is approximately Normally distributed with a mean of 7.7 grams and a standard deviation of 0.2 grams. Which of the following represents the probability that the mean weight of a random sample of 3 olives from this population is greater than 8 grams?
Understand the Problem
The question is asking for the probability that the mean weight of a sample of 3 olives from a normally distributed population exceeds 8 grams, with a given mean and standard deviation. To solve this, we would apply the concepts of the normal distribution and calculate the relevant probability.
Answer
The probability is approximately $0.0047$.
Answer for screen readers
The probability that the mean weight of a random sample of 3 olives is greater than 8 grams is approximately 0.0047.
Steps to Solve
- Define the Parameters of the Normal Distribution
The population of olive weights is normally distributed with:
- Mean ($\mu$) = 7.7 grams
- Standard deviation ($\sigma$) = 0.2 grams
- Calculate the Standard Error of the Sample Mean
The standard error (SE) for the sample mean of size $n = 3$ is calculated using the formula:
$$ SE = \frac{\sigma}{\sqrt{n}} = \frac{0.2}{\sqrt{3}} \approx 0.1155 \text{ grams} $$
- Determine the Z-score for the Desired Mean Weight
Next, we need to calculate the Z-score for a sample mean greater than 8 grams. The Z-score is calculated using the formula:
$$ Z = \frac{\bar{x} - \mu}{SE} = \frac{8 - 7.7}{0.1155} \approx 2.60 $$
- Find the Probability Using the Z-score
To find the probability that the mean weight is greater than 8 grams, we look up the Z-score in the standard normal distribution table or use a calculator to find the cumulative probability.
The cumulative probability for $Z = 2.60$ is approximately 0.9953, which means:
$$ P(Z > 2.60) = 1 - 0.9953 = 0.0047 $$
The probability that the mean weight of a random sample of 3 olives is greater than 8 grams is approximately 0.0047.
More Information
This low probability indicates that it's very unlikely for the average weight of a sample of 3 "Colossal" olives to exceed 8 grams, given the stated mean and standard deviation.
Tips
- Confusing standard deviation with standard error: The standard deviation applies to the population, while the standard error applies to the sample mean.
- Incorrect Z-score calculations: Ensure to subtract the mean correctly and divide by the standard error.
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