Respuesta :
Answer:
a) 68.98% probability that the mean life of a random sampleof 9 such machines falls between 6.4 and 7.2years
b) x = 7.346 years
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 normally distributed samples are 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, which is the area to the left of Z. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X, which is the area to the right of Z.
Central Limit Theorem
The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
In this problem, we have that:
[tex]\mu = 7, \sigma = 1, n = 9, s = \frac{1}{\sqrt{9}} = 0.3333[/tex]
(a) the probability that the mean life of a random sampleof 9 such machines falls between 6.4 and 7.2years;
This is the pvalue of Z when X = 7.2 subtracted by the pvalue of Z when X = 6.4. So
X = 7.2
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{7.2 - 7}{0.3333}[/tex]
[tex]Z = 0.6[/tex]
[tex]Z = 0.6[/tex] has a pvalue of 0.7257
X = 6.4
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{6.4 - 7}{0.3333}[/tex]
[tex]Z = -1.8[/tex]
[tex]Z = -1.8[/tex] has a pvalue of 0.0359
0.7257 - 0.0359 = 0.6898
68.98% probability that the mean life of a random sampleof 9 such machines falls between 6.4 and 7.2years
(b) the value of x to the right of which 15% of themeans computed from random samples of size 9would fall.
This is the 100-15 = 85th percentile, which is X when Z has a pvalue of 0.85. So it is X when Z = 1.037.
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]1.037 = \frac{X - 7}{0.3333}[/tex]
[tex]X - 7 = 1.037*0.3333[/tex]
[tex]X = 7.346[/tex]
The probability that the mean life of a random sample of 9 such machines falls between 6.4 and 7.2 years is 68.98%
What is z score?
Z score is used to determine by how many standard deviations the raw score is above or below the mean.
It is given by:
z = (raw score - mean) / (standard deviation÷√sample)
Mean = 7, standard deviation = 1, sample = 9.
For x = 6.4:
z = (6.4 - 7)/ (1 ÷√9) = -1.8
For x = 7.2:
z = (7.2 - 7)/ (1 ÷√9) = 0.6
P(-1.8 < z < 0.6) = P(z < 0.6) - P(z < -1.8) = 0.7257 - 0.0359 = 0.6898
The probability that the mean life of a random sample of 9 such machines falls between 6.4 and 7.2 years is 68.98%
Find out more on z score at: https://brainly.com/question/25638875
