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Let's break down the problem.

We have the following information:

- The machine produces defective pistons 1% of the time (probability \( p = 0.01 \)).
- A random sample of 90 pistons is taken (so \( n = 90 \)).

This problem involves a **binomial distribution**, where:
- \( n \) is the number of trials (pistons tested),
- \( p \) is the probability of success (a defective piston),
- The expectation (mean) of a binomial distribution is \( \mu = n \cdot p \),
- The standard deviation is \( \sigma = \sqrt{n \cdot p \cdot (1 - p)} \).

### Part (a): Estimate the number of defective pistons (mean of the distribution)
\[
\mu = n \cdot p = 90 \cdot 0.01 = 0.9
\]
So, the estimated number of defective pistons is **0.9**.

### Part (b): Standard deviation of the distribution
\[
\sigma = \sqrt{n \cdot p \cdot (1 - p)} = \sqrt{90 \cdot 0.01 \cdot 0.99} = \sqrt{0.891} \approx 0.944
\]
So, the standard deviation is approximately **0.944**.

Do you need further details or have any questions?

Here are 5 related questions to expand on this:

1. How does the size of the sample affect the standard deviation in a binomial distribution?
2. What would happen to the expected number of defective pistons if the defect rate doubled to 2%?
3. Can you use a normal approximation to the binomial in this case? Why or why not?
4. How would the calculations change if the sample size increased to 200 pistons?
5. What does the standard deviation tell us about the variability of defective pistons in this sample?

**Tip**: For large sample sizes and small probabilities, the binomial distribution can often be approximated by a normal distribution.

A machine that manufactures automobile pistons is estimated to produce a defective piston 1% of the time. Suppose that this estimate is correct and that a random sample of 90 pistons produced by this machine is taken. Estimate the number of defective pistons in the sample, and quantify the uncertainty of the estimate by giving the standard deviation.

Learn how to estimate the number of defective pistons produced by a machine using a binomial distribution. This question involves calculating the mean and standard deviation for a sample size of 90 pistons with a defect rate of 1%. Suitable for students in Grades 10-12.

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