An insurance company is reviewing its current policy rates based on the average

Question

An insurance company is reviewing its current policy rates based on the average claim amount mu. They randomly selected a sample of 40 claims, and found a sample mean $1950. Assuming that the standard deviation of claims is $250. 1) At a confidence level 95 %, find the margin of error for the population mean mu. Then, construct a 95 % confidence interval for mu. 2) For this sample of 40 claims with $1950 and a standard deviation of $250, the company wants to construct a test hypothesis to determine if the average claim amount is less than mu_0 = $1970 At a significance level 0.01, can we conclude that the average claim amount is less than mu_0 = 1970 $. a) Formulate the null and alternative hypothesis. b) At a significance level of alpha = 0.01, give your conclusion.

Explanation / Answer

Solution1:

z alpha/2 for 95%=1.96

Margin of error=z alpha/2(sigma/sqrt(n)

=1.96(250)/sqrt(40)

=77.475

95% Ci for population mean is

sample mean-margin of error<mu<sample mean+margin of error

1950-77.475<mu<1950+77.475

1872.52<mu<2027.48

we are 95% confident that the true population average of cliam amount lies in between 1872.52 and 2027.48

Solution2:

Null hypothesis:

H0: mean=1970

Alternative hypothesis

H1:mean<1970

Solution2b:

alpha=0.01

z=1950-1970/250/sqrt(40)

=-0.51

The P-Value is 0.305026.

The result is not significant at p < 0.01.

Decsion rule:

if p<0.01 reject null hypothesis

if p>0.01 fail to reject null hypothesis

Fail to reject Null hypothesis

Conclusion:

there is no suffiicent evidence at 5% level of significance that avaerage claim amount is less than $1970

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