Q: A sample of $$n=36$$ has a sample mean $$\bar{x} = 50$$ and sample standard deviation $$s = 6. $$What is the $$95\%$$ confidence interval for the population mean?

$$50 \pm 1.96 \times \frac{6}{\sqrt{36}}$$

Question 1

Which of the following best describes the main purpose of using statistics in engineering practice?

Accepted Answer

To analyze data, solve engineering problems, and support decision-making in real-world applications.

Question 2

Suppose the probability of a machine failing on any given day is $$0.02. $$What is the probability that the machine will not fail on a given day?

Accepted Answer

$$0.98$$

Question 3

Which of the following is a property of a probability distribution for a discrete random variable $$X$$?

Accepted Answer

Both $$0 \leq P(X = x_i) \leq 1$$ for all $$x_i$$ and $$\sum_{i} P(X = x_i) = 1$$

Question 4

A fair die is rolled. What is the expected value of the outcome?

Accepted Answer

$$3.5$$

Question 5

Which of the following distributions is most appropriate for modeling the number of emails received per hour?

Accepted Answer

Poisson distribution

Question 6

If $$X is a $$continuous random variable with probability density function $$f(x)$$, which equation must always be true?

Accepted Answer

$$\int_{-\infty}^{\infty} f(x) dx = 1$$

Question 7

A light bulb has a lifetime that is exponentially distributed with mean $$1000$$ hours. What is the probability that a randomly selected bulb lasts more than $$2000$$ hours?

Accepted Answer

$$e^{-2}$$

Question 8

Which of the following is NOT a valid assumption for using the binomial distribution?

Accepted Answer

The number of trials is infinite.

Question 9

A sample of $$n=36$$ has a sample mean $$\bar{x} = 50$$ and sample standard deviation $$s = 6. $$What is the $$95\%$$ confidence interval for the population mean?

Accepted Answer

$$50 \pm 1.96 	imes \frac{6}{\sqrt{36}}$$

Question 10

Which of the following statements about the normal distribution is TRUE?

Accepted Answer

It is completely described by its mean and variance.

Question 11

A hypothesis test is performed at the $$\alpha = 0.05$$ significance level. The $$p-$$value is $$0.03. $$What is the correct conclusion?

Accepted Answer

Reject the null hypothesis.

Question 12

Which formula gives the expected value $$E[X]$$ for a discrete random variable $$X$$ with probability mass function $$p(x)$$?

Accepted Answer

$$E[X] = \sum_{x} x \cdot p(x)$$

Question 13

A quality control engineer wants to test if the proportion of defective items in a batch is greater than $$0.05. $$Which hypothesis test should be used?

Accepted Answer

A one-tailed test for proportions

Question 14

Which of the following is the formula for the variance of a discrete random variable $$X$$?

Accepted Answer

$$Var(X) = E[X^2$] - (E[X])^2$

Question 15

Which distribution would be most appropriate for modeling the time between arrivals of buses at a station, assuming arrivals are random?

Accepted Answer

Exponential distribution

Question 16

A sample of $$n=25 is $$drawn from a normal population with unknown variance. Which distribution should be used to construct a confidence interval for the mean?

Accepted Answer

$$t-$$distribution

Question 17

Which of the following is a key assumption for the use of the Central Limit Theorem?

Accepted Answer

The sample size is large enough.

Question 18

In a hypothesis test, what is the probability of making a Type I error?

Accepted Answer

$$\alpha$$

Question 19

Which of the following is the correct formula for the probability mass function of a binomial random variable $$X$$ with parameters $$n$$ and $$p$$?

Accepted Answer

$$P(X = k) = \binom{n}{k} p^k (1-p$$)^{n-k}$$

IE 3302 Engineering Statistics: Syllabus and Course Structure Study Notes

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