Question: Everyone is feeling the pinch of inflated living cost due to inflation in 2023-2024. The average rent for a two-bed room apartment in Ottawa is$1,800 per month with a standard deviation of$300 in January 2024. The distribution of the house rent for two-bed room apartment in Ottawa in 2024 January is assumed to follow Normal distribution. LetX be the random variable that represent the monthly rent for a two-bed room apartment in Ottawa .

1. If a random sample of 10 two-bed room apartments is taken in 2024 January, what is the probability that the average monthly rent exceeds$2,000? What assumptions are required for the validity of this result?
2. Which range of average monthly rent covers middle 80% for a random sample of 10 two-bed room apartments?
3. If a random sample of 35 two-bed room apartments is taken in 2024 January, what is the probability that the average monthly rent is at most$1,850?
4. One of your Telfer classmates warns you that the distribution of monthly rent for two-bed room apartment in Ottawa is a left-skewed distribution and asks you not to trust the assumption of normality. Would your answers in (a) through (c) be still valid if you consider your classmate's suggestion? Justify your answer.
5. Suppose you have learned that 35% apartment rents in January 2024 are more than$2,000. What is the approximate probability that at most 30 apartment rents are more than$2,000 in a random sample of 100 apartments? What assumptions are required for the validity of this result? (Hint: Sampling Distribution of Proportions)