The accounting at GEICO notices that the average claim might be larger than $180

Question

The accounting at GEICO notices that the average claim might be larger than $1800, which was previously set as a true average claim amount. If the actual average is higher, the company wil be loosing money in the long run. They ran- domly select 45 claims, and calculate a sample mean of $1,920. Suppose that the variance of claims is known to be equal to $1600. (a) Set the significance level to = 0.05 and use the hypothesis testing to check if the average claim is larger than $1800. (b) Find the probability of Type I error. (c) Find the p-value corresponding to the given sample.

Explanation / Answer

Solution:

Given that, =1800, =40, x =1920
number (n)=45

Test Hypothesis:
Ho: =1800
H1: >1800

a) level of significance, = 0.05
from standard normal table,right tailed z /2 =1.645
since our test is right-tailed
reject Ho, if zo > 1.645
we use test statistic (z) = x-/(/sqrt(n))
zo = 1920-1800/(40/sqrt(45)) = 20.12
| zo | = 20.12
critical value
the value of |z| at los 5% is 1.645
we got |zo| =20.12 & | z | = 1.645
make decision
hence value of | zo | > | z | and here we reject Ho
p-value : right tail - ha : ( p > 20.12 ) = 0
hence value of p0.05 > 0, here we reject Ho
Conclude: There is evidence that the average claim is larger than $1800.

b) P(Type I error is just the p value we got earlier, which is pretty close to 0)

c) p-value : right tail - ha : ( p > 20.12 ) = 0
hence value of p0.05 > 0, here we reject Ho
P value to given sample is same as Type I error, so its about 0. There is a possibility that they wanted you to write Type I error as 0.05, which was given, in which my answers above would change, but I don't think that was the case.

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