Answered: The lengths of nails manufactured at a…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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The lengths of nails manufactured at a factory are normally distributed with a mean length of 4.43 cm and a standard deviation of 1.41 cm. For quality control purposes, a random sample of 14 nails is chosen once per day, and measured. Let X be the random variable representing the mean length of the nails.

A. Fill in the blank, rounding your answers to 2 decimal places if needed. According to the Central Limit Theorem, X is approximately normal with a mean of _______cm and a standard error of the mean________cm

b. Find the z-score associated to a sample with a mean of 4.56 cm, using the sampling distribution. Round your answer to two decimal places. ______

c. Find the probability that a randomly selected sample of 14 nails has a mean length higher than 4.56 cm. Round your answer to 4 decimal places.________

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

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c. Find the probability that a randomly selected individual nail has a length higher than 4.56 cm. Round your answer to 4 decimal places.

e. The machine that cuts the nails will be pulled for inspection and repairs if the sample mean of the quality control sample is in the bottom 3% or top 3% of possible sample means. What sample average lengths will trigger a repair? Round your answers to 2 decimal places.

The machine will be pulled for repairs if the quality control sample has a mean length below_______ cm or above_______ cm.

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