# An online used car company sells second-hand cars. For 30 randomly selected tran

An online used car company sells second-hand cars. For 30 randomly selected transactions, the mean price is 2800 dollars.
a) Assuming a population standard deviation transaction prices of 130 dollars, obtain a $$\displaystyle{99}\%$$ confidence interval for the mean price of all transactions.
b) Which of the following is the correct interpretation for your answer in part (a)?
A) If we repeat the study many times, $$\displaystyle{99}\%$$ of the calculated confidence intervals will contain the mean price of all transactions.
B) There is a $$\displaystyle{99}\%$$ change that the mean price of all transactions lies in the interval
C) We can be $$\displaystyle{99}\%$$ confident that the mean price for this sample of 30 transactions lies in the interval
D) None of the above

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Step 1
The provided information are:
Sample size $$\displaystyle{\left({n}\right)}={30}$$
Sample mean $$\displaystyle{\left(\overline{{{x}}}\right)}={2800}$$
Population standard deviation $$\displaystyle{\left(\sigma\right)}={130}$$
Confidence level $$\displaystyle={99}\%$$
Step 2
a) Using the standard normal table, the z-critical value at $$\displaystyle{99}\%$$ confidence level is 2.58.
The confidence interval is:
$$\displaystyle{C}{I}=\overline{{{x}}}\pm{z}\times{\frac{{\sigma}}{{\sqrt{{{n}}}}}}$$
$$\displaystyle={2800}\pm{2.58}\times{\frac{{{130}}}{{\sqrt{{{30}}}}}}$$
$$\displaystyle={\left({2738.9},\ {2861.1}\right)}$$
Thus, the $$\displaystyle{99}\%$$ confidence interval is $$\displaystyle{\left({2738.9},\ {2861.1}\right)}$$
Step 3
b) If many confidence interval of same size $$\displaystyle={\left({30}\right)}$$ is taken and confidence interval is calculated for each of the sample, then about $$\displaystyle{99}\%$$ of them will contain the true population parameter.
Thus, the correct option is (A).