A university reports that $3.2$ percent of its students are international students. If a random sample of $900$ students is selected and the proportion of international students in the sample is calculated, what is the mean of the sampling distribution of the sample proportions?

A university has $4200$ professors, and their research grant amounts are distributed in a highly skewed manner. If random samples of $36$ professors are repeatedly selected and the mean grant amount is calculated for each sample, what is the approximate shape of the distribution of these sample means?

In a biology study, a population consists of three blue butterflies and two orange butterflies. Two butterflies are randomly chosen with replacement. What is the sampling distribution of the proportion of blue butterflies in the samples?

A company produces a new type of LED bulb and claims that $80\%$ of the bulbs last more than $10,000$ hours. To verify the claim, a random sample of $120$ bulbs is tested. The quality assurance team decides that if fewer than $90$ bulbs pass the test, they will reject the company’s claim. What is the probability that the claim will be rejected, assuming the company's claim is true?

A nutritionist is studying the heights of female volleyball players. She records the heights (in inches) of $5$ randomly selected players: $68,70,65,69,72$. What is the sample standard deviation of these heights?