Answer:
There is a 99.24% probability that Claude's sample has a mean between 119.985 and 120.0125 inches.
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:
The population of rods has a mean length of 120 inches and a standard deviation of 0.05 inch. This means that .
Claude Ong, manager of Quality Assurance, directs his crew measure the lengths of 100 randomly selected rods. This means that .
The probability that Claude's sample has a mean between 119.985 and 120.0125 inches is
We are working with the sample mean, so we use the standard deviation of the sample, that is, instead of
in the z score formula.
This probability is the pvalue of Z when subtracted by the pvalue of Z when
.
X = 120.0125
has a pvalue of 0.9938.
X = 119.985
has a pvalue of 0.0014.
So there is a 0.9938 - 0.0014 = 0.9924 = 99.24% probability that Claude's sample has a mean between 119.985 and 120.0125 inches.