Answer and Explanation:
1. To construct a 95% confidence interval for the average monthly premiums for owners of medium sized vehicles, we need to know the sample mean, sample size, and population standard deviation. In this case, the sample mean is R550, the sample size is 250, and the population standard deviation is not given. Without the population standard deviation, we cannot calculate the standard error of the mean and therefore cannot construct a confidence interval.
2. If the sample size increases to 500 car owners and the population standard deviation becomes R58, we can estimate the 90% confidence interval for the total monthly premium of the company. The standard error of the mean is calculated as `SE = s / sqrt(n)`, where `s` is the population standard deviation and `n` is the sample size. In this case, `SE = 58 / sqrt(500) = 2.594`.
To calculate a 90% confidence interval, we need to find the critical value for a t-distribution with `n - 1` degrees of freedom. For a sample size of 500, this would be a t-distribution with 499 degrees of freedom. Using a t-table or calculator, we can find that the critical value for a 90% confidence interval with 499 degrees of freedom is approximately 1.645.
The margin of error for the confidence interval is calculated as `ME = critical value * SE`. In this case, `ME = 1.645 * 2.594 = 4.265`. Therefore, the 90% confidence interval for the total monthly premium of the company is `(550 - ME, 550 + ME)`, or `(545.735, 554.265)`.
This confidence interval can be interpreted as follows: We are 90% confident that the true average monthly premium for owners of medium sized vehicles is between R545.735 and R554.265.