Answer:
P(215<X<225) = 0.7620
Step-by-step explanation:
The shape of the distribution is unknow, however, the shape of the sampling distributions of the sample mean is approximately normal due to the central limit theorem.
Normal probability distribution
Problems of normally distributed samples are 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.
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.
In this problem, we have that:
P(215<X<225)
This is the pvalue of Z when X = 225 subtracted by the pvalue of Z when X = 215. So
X = 225
By the Central Limit Theorem
has a pvalue of 0.8810
X = 215
has a pvalue of 0.1190
0.8810 - 0.1190 = 0.7620
P(215<X<225) = 0.7620
