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Question
1)
"The diameter of a wire should be 8 cm. It follows a normal distribution and has a standard deviation of 0.01 cm.
A person tests each delivery by taking 5 samples, accepting the goods only if the mean is at most 8.01
a)
Find the probability of type 1 error?
b)
A hypothesis test is to be carried out with P(type I error) = P(type II error) and either one being at 0.1%.
What is the necessary sample size for this test? (answer in my book says it's n=17)
2)
A company will sell a new product if the proportion of customers that will buy it is at least 20%. It is decided that the null hypothesis is rejected if the sample proportion is at least 27%. Sample size = 100.
a) Find P(type I error)
b)
If P(do not reject the null hypothesis)must be at least 99% if the true proportion is 0.17 and
P(reject the null hypothesis)must be at least 99% if the true proportion is 0.27:
which sample size is required?
(My book's answer is n = or greater than 365)
I don't understand how to solve these kind of questions, I'd greatly appreciate if you could help explain !
1a) The chance of a type I error is P(z>µ+kφ).
In this case, µ = 8 and σ = √(0.01²/5) = √0.00002 = 0.0045.
Note that the maximum error is 0.01 and σ = 0.0045.
If we take z=0.01/0.0045, we get the the number of standard deviations
to look up on a Standard Normal Distribution Table.
Be sure and pay attention to where z lies to give that value on the Normal Table looked at.
1b) We need to know what the true average is for a type II error to occur.
2a) Given the average is 20%, σ = √(0.2*(1-0.2)/n).
The probability of a type I error is found by noting what value is gotten from z (0.27-0.20)/σ.
2b) Let z = (0.2-0.17)/√(0.17(1-0.17)/n); figure the chance of it being less than this value
using Standard Norml Table; it must be at least 0.99.
Take z = (0.27-0.2)/√(0.27(1-0.27)/n); figure the chance of it being greater than this value
using the Standard Normal Table; it must be at least 0.99.
Basically, the value of n must be large enough so that the σ value is small enough to make the appropriate value on the z table look up as 0.99.
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