Answer:
There is a 28.81% probability that they have a mean cost between 267 dollars and 269 dollars.
Step-by-step explanation:
The Central Limit Theorem estabilishes that, for a random variable X, with mean and standard deviation , a large sample size can be approximated to a normal distribution with mean and standard deviation
Problems of normally distributed samples can be solved using the z-score formula.
In a set with mean and standard deviation , the zscore of a measure X is given by:
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.
In this problem, we have that:
.
Find the probability that they have a mean cost between 267 dollars and 269 dollars.
This probability is the pvalue of Z when X = 269 subtracted by the pvalue of Z when X = 267. So:
X = 269
has a pvalue of 0.7881.
X = 267
has a pvalue of 0.50.
So there is a 0.7881 - 0.50 = 0.2881 = 28.81% probability that they have a mean cost between 267 dollars and 269 dollars.