Answer:
The probability is 0.789
Step-by-step explanation:
To solve this question, we need to understand the normal probability distribution and 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:
Find the probability that the mean age of the individuals sampled is within one year of the mean age for all individuals in the city.
Using , this is the pvalue of Z when X = 31 subtracted by the pvalue of Z when X = 29. So
X = 31
By the Central Limit Theorem
has a pvalue of 0.8944
X = 29
has a pvalue of 0.1056
0.8944 - 0.1056 = 0.7888
Rounded to three decimal places, 0.789
The probability is 0.789