Answer:
a)
b)
And solving for c we got:
So then the value of c = 5.88
c) Let X the niumber of acceptable specimens among a sampel of 10. The acceptable range from part a) is between 64 and 78 and we found that that would represent our p value, and we select samples of size n =10, so then the expected value would be:
d)
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 hardness 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 with this difference:
And in order to find these probabilities we can find tables for the normal standard distribution, excel or a calculator.
Part b
For this case we want to find the value of c who satisfy this condition:
For this case we can use the z score formula, we can find two values on the normal dtandard distribution that accumulates 0.95 of the values on the middle and for this case the two values are:
So using the z score formula we have this:
And solving for c we got:
And solving for c we got:
So then the value of c = 5.88
Part c
Let X the niumber of acceptable specimens among a sampel of 10. The acceptable range from part a) is between 64 and 78 and we found that that would represent our p value, and we select samples of size n =10, so then the expected value would be:
Part d
For this case we define the random variable Y= the number among the ten specimens with hardness less than 73.52, for this case we can find the probability like this:
[te]x P(X<73.52) = P(Z< \frac{73.52-71}{3}=P(Z<0.84)=0.800[/tex]
So then we can define our random variable like this:
And we want to find this probability:
The probability mass function for the Binomial distribution is given as:
For this case we can use the complement rule: