Respuesta :
Answer:
On the first day, 98.75% probability that the sample average amount of time taken is at most 11 min.
On the second day, 99.29% probability that the sample average amount of time taken is at most 11 min.
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. 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 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 = 8, \sigma = 3[/tex]
First day
[tex]n = 5, s = \frac{3}{\sqrt{5}} = 1.34[/tex]
The probability is the pvalue of Z when X = 11. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{11 - 8}{1.34}[/tex]
[tex]Z = 2.24[/tex]
[tex]Z = 2.24[/tex] has a pvalue of 0.9875
On the first day, 98.75% probability that the sample average amount of time taken is at most 11 min.
Second day
[tex]n = 6, s = \frac{3}{\sqrt{5}} = 1.2247[/tex]
The probability is the pvalue of Z when X = 11. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{11 - 8}{1.2247}[/tex]
[tex]Z = 2.45[/tex]
[tex]Z = 2.45[/tex] has a pvalue of 0.9929
On the second day, 99.29% probability that the sample average amount of time taken is at most 11 min.
