Calculate the mean 𝑛𝑝 and variance 𝑛𝑝(1 − 𝑝) of a binomial distribution using technology and algebraic methods.

A manufacturer tests batches of light bulbs for defects. Historical data shows that 8% of bulbs are defective. In a quality assurance sample of 250 bulbs, calculate the standard deviation of the number of defective bulbs.

A wildlife biologist is monitoring nesting success in a penguin colony. From long-term records, each pair has a probability of $p = 0.72$ of successfully raising at least one chick each season. The biologist selects $n = 50$ randomly chosen pairs for the current season. Let $X$ be the binomial random variable representing the number of successful pairs. Calculate: (a) the mean of $X$ [1 mark] (b) the standard deviation of $X$, correct to 2 decimal places [1 mark] (c) the variance of $X$, in exact form or to 2 decimal places [1 mark]

A Queensland wildlife sanctuary monitors the daily breeding behaviour of a colony of endangered frogs. Historical data shows that on any given day during the breeding season, the probability that at least one successful mating call is recorded is \(0.72\). The sanctuary conducts observations for \(50\) consecutive days during the breeding season. Let \(Y\) be the binomial random variable representing the number of days on which at least one successful mating call is recorded. (a) Calculate the mean and variance of \(Y\) using algebraic methods. (2 marks) (b) At the end of the \(50\)-day period, the sanctuary recorded successful calls on \(32\) days. Calculate how many standard deviations this observed value is from the mean. Give your answer correct to two decimal places. (2 marks)

A quality control inspector examines batches of electronic components. Historical data shows that 8% of components are defective. The inspector randomly selects 45 components from a large shipment for testing. Let \( X \) represent the number of defective components in the sample. Calculate the mean and variance of \( X \).