Q: Suppose $$X \sim N(40, 36)$$, where $$\mu = 40$$ and $$\$$sigma^2$$ = 36. $$What is the 95th percentile of $$X$$?

$$x_{0.95} = 40 + 6 \times 1.645 = 49.87$$

Q: If $$X \sim N(40, 36)$$, what is the 5th percentile of $$X$$?

$$x_{0.05} = 40 + 6 \times (-1.645) = 30.13$$

Q: Given a binomial experiment with $$n = 40$$ and $$p = 0.5$$, what is the standard deviation for the normal approximation?

$$\sigma = \sqrt{40 \times 0.5 \times 0.5} = \sqrt{10}$$

Question 1

Suppose $$X \sim N(40, 36)$$, where $$\mu = 40$$ and $$\$$sigma^2$$ = 36. $$What is the 95th percentile of $$X$$?

Accepted Answer

$$x_{0.95} = 40 + 6 	imes 1.645 = 49.87$$

Question 2

A binomial experiment is characterized by which of the following properties?

Accepted Answer

Each trial is independent, has two mutually exclusive outcomes, and the probability of success is constant.

Question 3

Given a binomial experiment with $$n = 500$$ and $$p = 0.07$$, what is the mean and standard deviation for the normal approximation?

Accepted Answer

$$\mu = 35$$, $$\sigma = 5.71$$

Question 4

Why is a continuity correction applied when using the normal approximation for a binomial probability?

Accepted Answer

To adjust for the difference between discrete and continuous distributions.

Question 5

In the automotive example, if $$n = 400$$ and $$p = 0.35$$, what is the value of $$z$$ used to approximate $$P(X \u003c 150)$$?

Accepted Answer

$$z = \frac{149.5 - 140}{9.54}$$

Question 6

Which condition must be satisfied to use the normal approximation for a binomial distribution?

Accepted Answer

$$np(1-p) \geq 10$$

Question 7

If $$X \sim N(40, 36)$$, what is the 5th percentile of $$X$$?

Accepted Answer

$$x_{0.05} = 40 + 6 	imes (-1.645) = 30.13$$

Question 8

A sample statistic is best described as:

Accepted Answer

A known but variable number computed from a sample.

Question 9

In the context of parameter estimation, what does the margin of error represent?

Accepted Answer

The range within which the true population parameter is likely to fall, given a certain confidence level.

Question 10

Given a binomial experiment with $$n = 40$$ and $$p = 0.5$$, what is the standard deviation for the normal approximation?

Accepted Answer

$$\sigma = \sqrt{40 	imes 0.5 	imes 0.5} = \sqrt{10}$$

Question 11

Which of the following is NOT a step in constructing a sampling distribution for a statistic?

Accepted Answer

Increase the population size until the sample mean equals the population mean.

Question 12

For a binomial random variable $$X$$ with $$n = 500$$ and $$p = 0.07$$, what is the $$z-$$score for $$x = 29.5$$ when using the normal approximation?

Accepted Answer

$$z = \frac{29.5 - 35}{\sqrt{32.55}}$$

Question 13

Which statement about the sampling distribution of the sample mean $$\bar{X} is $$correct?

Accepted Answer

It is the probability distribution of all possible values of $$\bar{X}$$ computed from samples of size $$n$$ from a population.

Question 14

In the context of normal approximation to the binomial, how would you approximate $$P(X = 18)$$ for $$n = 40$$, $$p = 0.5$$?

Accepted Answer

$$P(17.5 \u003c X \u003c 18.5)$$

Question 15

A survey reports that 41% of the public believes young adults are having the toughest time in today's economy, with a margin of error of $$\pm 2.9\% at 95$$% confidence. What is the correct interpretation?

Accepted Answer

We are 95% confident that between 38.1% and 43.9% of the public believe young adults are having the toughest time.

Question 16

Which of the following best describes a parameter?

Accepted Answer

A number that reflects properties of a population and is typically unknown.

Question 17

If you have a population $$S = \{0, 6, 9\}$$ and take samples of size $$n = 2$$ with replacement, what is the probability of obtaining a sample mean of $$3$$?

Accepted Answer

$$2/9$$

Question 18

Which factor does NOT affect the shape of the sampling distribution of a statistic?

Accepted Answer

The color of the data points

Question 19

In a binomial experiment with $$n = 400$$ and $$p = 0.35$$, what is the probability that at least 160 households have three or more cars?

Accepted Answer

$$P(Z \geq \frac{159.5 - 140}{9.54}) = 0.0207$$

Lecture 23: Normal Distributions, Binomial Approximations, and Parameter Estimation

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