Question

According to the National Coalition on Health Care, the mean annual premium for an employer health plan covering a family of four cost $11,500 in 2007. A random sample of 100 families of four taken this year showed a mean annual premium of $11,750. Assuming  = $1500 , test whether the mean annual premium has increased, using a 0.05 significant level. Use the p-value method.

Answer 1

Given that,
population mean(u)=11500
standard deviation, σ =1500
sample mean, x =11750
number (n)=100
null, Ho: μ=11500
alternate, H1: μ>11500
level of significance, α = 0.05
from standard normal table,right tailed z α/2 =1.645
since our test is right-tailed
reject Ho, if zo > 1.645
we use test statistic (z) = x-u/(s.d/sqrt(n))
zo = 11750-11500/(1500/sqrt(100)
zo = 1.667
| zo | = 1.667
critical value
the value of |z α| at los 5% is 1.645
we got |zo| =1.667 & | z α | = 1.645
make decision
hence value of | zo | > | z α| and here we reject Ho
p-value : right tail - ha : ( p > 1.667 ) = 0.048
hence value of p0.05 > 0.048, here we reject Ho
ANSWERS
---------------
null, Ho: μ=11500
alternate, H1: μ>11500
test statistic: 1.667
critical value: 1.645
decision: reject Ho
p-value: 0.048
we have enough evidence to support the claim that whether the mean annual premium has increased.