Respuesta :
Answer:
If 24 items are picked at random, 8% of the time their mean life will be less than 5.156 years.
Step-by-step explanation:
To solve this question, we have 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. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
Central limit theorem:
The Central Limit Theorem estabilishes that, for a random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sample means with size n of at least 30 can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex]
In this problem, we have that:
[tex]\mu = 5.5, \sigma = 1.2, n = 24, s = \frac{1.2}{\sqrt{24}} = 0.2449[/tex]
If 24 items are picked at random, 8% of the time their mean life will be less than how many years?
This is the value of X when Z has a pvalue of 0.08. So it is X when Z = -1.405.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]-1.405 = \frac{X - 5.5}{0.2449}[/tex]
[tex]X - 5.5 = -1.405*0.2449[/tex]
[tex]X = 5.156[/tex]
If 24 items are picked at random, 8% of the time their mean life will be less than 5.156 years.
