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Answer:
64.76% probability that the mean weight of the sample babies would differ from the population mean by less than 40 grams
Step-by-step explanation:
To solve this question, we have 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 random variable X, with mean and standard deviation
, a large sample size can be approximated to a normal distribution with mean
and standard deviation
In this problem, we have that:
What is the probability that the mean weight of the sample babies would differ from the population mean by less than 40 grams
This is the pvalue of Z when X = 3242 + 40 = 3282 subtracted by the pvalue of Z when 3242 - 40 = 3202
X = 3282
By the Central Limit Theorem
has a pvalue of 0.8238.
X = 3202
has a pvalue of 0.1762
0.8238 - 0.1762 = 0.6476
64.76% probability that the mean weight of the sample babies would differ from the population mean by less than 40 grams