Answer:
d. 0.0047
Step-by-step explanation:
To solve this question, we need to understand the normal probability distribution and the central limit theorem.
Normal probability distribution
When the distribution is normal, we use the z-score formula.
In a set with mean
and standard deviation
, the zscore of a measure X is given by:
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
Central Limit Theorem
The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean
and standard deviation
, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
and standard deviation
.
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
Population:
We have that ![\mu = 7.7, \sigma = 0.2](https://tex.z-dn.net/?f=%5Cmu%20%3D%207.7%2C%20%5Csigma%20%3D%200.2)
Sample of 3:
![n = 3, s = \frac{0.2}{\sqrt{3}} = 0.1155](https://tex.z-dn.net/?f=n%20%3D%203%2C%20s%20%3D%20%5Cfrac%7B0.2%7D%7B%5Csqrt%7B3%7D%7D%20%3D%200.1155)
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?
This is 1 subtracted by the pvalue of Z when X = 8. So
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
By the Central Limit Theorem
![Z = \frac{X - \mu}{s}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7Bs%7D)
![Z = \frac{8 - 7.7}{0.1155}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7B8%20-%207.7%7D%7B0.1155%7D)
![Z = 2.6](https://tex.z-dn.net/?f=Z%20%3D%202.6)
has a pvalue of 0.9953
1 - 0.9953 = 0.0047
The probability is given by option d.