Choose all the numbers that are multiples of 7. 7 28 34 49 21 54 63
1 answer:
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Answer:
a)
And we can find this probability using the z table or excel:
b)
c) And since the distribution of X is normal we can conclude that the distribution for the sample mean is also normal and for this reason we can use the z table no matter if the sample size is small.
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".
Part a
Let X the random variable that represent the interpupillary distance of a population, and for this case we know the distribution for X is given by:
Where and
We are interested on this probability
And the best way to solve this problem is using the normal standard distribution and the z score given by:
If we apply this formula to our probability we got this:
And we can find this probability using the z table or excel:
Part b
For this case the distirbution of the sample mean is also normal since the distribution for the random variable X is normal and is given by:
And for this case we want this probability:
And we can find this probability on this way:
And in order to find these probabilities we can find tables for the normal standard distribution, excel or a calculator.
Part c
For this case since the sample mean is defined as:
If we find the expected value for this variable we got:
And for the variance we have:
And for this reason the deviation is
And since the distribution of X is normal we can conclude that the distribution for the sample mean is also normal and for this reason we can use the z table no matter if the sample size is small.