Respuesta :
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 [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], a large sample size can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]\frac{\sigma}{\sqrt{n}}[/tex].
Problems of normally distributed samples can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
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 [tex]\mu = 120, \sigma = 0.05[/tex].
Claude Ong, manager of Quality Assurance, directs his crew measure the lengths of 100 randomly selected rods. This means that [tex]n = 100, s = \frac{\sigma}{\sqrt{n}} = \frac{0.05}{\sqrt{100}} = 0.005[/tex].
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, [tex]s[/tex] instead of [tex]\sigma[/tex] in the z score formula.
This probability is the pvalue of Z when [tex]X = 120.0125[/tex] subtracted by the pvalue of Z when [tex]X = 119.985[/tex].
X = 120.0125
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{120.0125 - 120}{0.005}[/tex]
[tex]Z = 2.5[/tex]
[tex]Z = 2.5[/tex] has a pvalue of 0.9938.
X = 119.985
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{119.985 - 120}{0.005}[/tex]
[tex]Z = -3[/tex]
[tex]Z = -3[/tex] 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.
