1. **Problem Statement:** We have a machine that produces defective pistons with probability $p=0.03$. A random sample of $n=90$ pistons is taken. We want to find: (a) The expected number of defective pistons (mean of the distribution). (b) The standard deviation of the number of defective pistons. 2. **Distribution and Formula:** The number of defective pistons in the sample follows a binomial distribution $X \sim \text{Binomial}(n, p)$ where $n=90$ and $p=0.03$. - The mean (expected value) of a binomial distribution is given by: $$\mu = np$$ - The standard deviation is given by: $$\sigma = \sqrt{np(1-p)}$$ 3. **Calculate the Mean:** $$\mu = 90 \times 0.03 = 2.7$$ This means on average, we expect 2.7 defective pistons in the sample. 4. **Calculate the Standard Deviation:** $$\sigma = \sqrt{90 \times 0.03 \times (1 - 0.03)} = \sqrt{90 \times 0.03 \times 0.97}$$ $$= \sqrt{2.619} \approx 1.619$$ 5. **Interpretation:** - The mean tells us the expected number of defective pistons. - The standard deviation quantifies the variability or uncertainty around this mean. **Final answers:** (a) Mean = $2.7$ (b) Standard deviation $\approx 1.619$ (rounded to three decimal places)