Answer:
The sample mean that will cut off the upper 95% of all samples of size 20 taken from the population is of 1963.2 pounds.
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 normal distributions can be solved using the z-score formula.
In a set with mean and standard deviation , the z-score 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 p-value, we get the probability that the value of the measure is greater than X.
Central Limit Theorem
The Central Limit Theorem establishes 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.
The average breaking strength of a certain brand of steel cable is 2000 pounds, with a standard deviation of 100 pounds.
This means that
A sample of 20 cables is selected and tested.
This means that
Find the sample mean that will cut off the upper 95% of all samples of size 20 taken from the population.
This is the 100 - 95 = 5th percentile, which is X when Z has a p-value of 0.05, so X when Z = -1.645. So
By the Central Limit Theorem
The sample mean that will cut off the upper 95% of all samples of size 20 taken from the population is of 1963.2 pounds.