An insurance company is reviewing its current policy rates.when originally setting the rates they believed that the average claim amount was 1800.00.they are concerned that the true mean is actually higher than this,because they could potentially lose a lot of money.they randomly select 40 claims,and calculate a sample mean of 1950.00.Assuming that the standard deviation of claims is 500.00,and set a=0.05,test to see if the insurance company should be concerned.
Z = (mean1 - mean2)/standard error (SE) of difference between means
SEdiff = √(SEmean1^2 + SEmean2^2)
SEm = SD/√n
If only one SD is provided, you can use just that to determine SEdiff.
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability of the Z score. Is it <.05?
To test if the insurance company should be concerned about the true mean claim amount being higher than their original belief of $1800, we can use a hypothesis test.
Step 1: Define the hypotheses.
The null hypothesis (H₀): The true mean claim amount is not higher than $1800.
The alternative hypothesis (H₁): The true mean claim amount is higher than $1800.
Step 2: Choose the significance level (α).
The significance level determines the probability of rejecting the null hypothesis when it is actually true. In this case, α is given as 0.05.
Step 3: Calculate the test statistic.
We'll calculate the test statistic which follows a t-distribution. The formula for the test statistic is:
t = (sample mean - population mean) / (sample standard deviation / √n)
Given:
sample mean (x̄) = $1950
population mean (μ) = $1800
sample standard deviation (σ) = $500
sample size (n) = 40
t = (1950 - 1800) / (500 / √40)
Step 4: Determine the critical value.
Since the alternative hypothesis is that the true mean is higher, we'll perform a one-tailed test. With a significance level of 0.05 and degrees of freedom (df) = n - 1, we can use a t-distribution table or a t-distribution calculator to find the critical value.
For this case, with df = 39 and α = 0.05, the critical value is approximately 1.684.
Step 5: Compare the test statistic and critical value.
If the test statistic (t-value) is greater than the critical value, we can reject the null hypothesis in favor of the alternative hypothesis. If the test statistic is less than or equal to the critical value, we fail to reject the null hypothesis.
In this case, the calculated t-value can be compared to the critical value.
Step 6: Calculate the p-value (optional).
If you prefer to use the p-value instead of comparing the test statistic to the critical value, you can calculate it using the t-distribution. The p-value represents the probability of obtaining a test statistic as extreme or more extreme than the observed statistic, assuming the null hypothesis is true.
Once the p-value is calculated, compare it to the significance level (α). If the p-value is less than α, reject the null hypothesis; otherwise, fail to reject the null hypothesis.
To calculate the p-value, we need to find the area under the t-distribution curve to the right of the calculated t-value (since we have a one-tailed test).
Note: Depending on the programming language or statistical software you are using, you can either use built-in functions or t-distribution tables to calculate the p-value.
Step 7: Make a conclusion.
Based on the comparison of the test statistic and the critical value or p-value, make a conclusion about whether to reject or fail to reject the null hypothesis.
In this case, compare the calculated t-value to the critical value of 1.684. If the calculated t-value is greater than 1.684, reject the null hypothesis and conclude that the true mean claim amount is indeed higher than $1800. If the calculated t-value is less than or equal to 1.684, fail to reject the null hypothesis and conclude that there is not enough evidence to suggest that the true mean claim amount is higher than $1800.
Note: The actual calculations and conclusion will depend on the exact values obtained when calculating the test statistic and critical value.