Answer:
a) For this case we select a sample size of n=9. And we know that the distribution of X is normal so then the distribution for the sample mean is given by:
b)
c)
And we can use the z score given by:
And if we find the z score for 39.5 we got:
And using the complement rule we got:
d)
And we can use the z score given by:
And if we find the z score for 37.5 we got:
And using the complement rule we got:
Step-by-step explanation:
Previous concepts
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".
Solution to the problem
Let X the random variable that represent the life of batteries of a population, and for this case we know the distribution for X is given by:
Where and
Part a
For this case we select a sample size of n=9. And we know that the distribution of X is normal so then the distribution for the sample mean is given by:
Part b
The standard error is given by:
Part c
We want this probability:
And we can use the z score given by:
And if we find the z score for 39.5 we got:
And using the complement rule we got:
Part d
We want this probability:
And we can use the z score given by:
And if we find the z score for 37.5 we got:
And using the complement rule we got: