Q: An insurance company claims that 75% of policyholders have smoke detectors. In a random sample of 80 applications, what is the standard deviation of the sample proportion of clients with smoke detectors?

$$\sqrt{\frac{0.75 \times 0.25}{80}}$$

Question 1

Which of the following is NOT a required condition for a random variable to follow a Binomial distribution?

Accepted Answer

Each trial must have a different probability of success.

Question 2

Suppose you attend a speed dating event and meet 10 people. If the probability of liking each person is $$0.1$$, what is the expected number of people you will like?

Accepted Answer

$$1$$

Question 3

A Binomial random variable $$X$$ with parameters $$n=1000$$ and $$p=0.1$$ can be approximated by a Normal distribution if which of the following conditions is satisfied?

Accepted Answer

$$np \u003e 10$$ and $$n(1-p) \u003e 10$$

Question 4

Given $$X \sim 	ext{Binomial}(n, p)$$, what are the mean and variance of the Normal distribution used to approximate $$X$$ when $$n is $$large?

Accepted Answer

Mean $$= np$$, Variance $$= np(1-p)$$

Question 5

Why is the Normal approximation to the Binomial distribution useful in practice?

Accepted Answer

It simplifies probability calculations when $$n is $$large.

Question 6

Suppose you look through 1000 online profiles per day for 365 days, and the probability of liking each is $$0.1. $$What is the approximate shape of the distribution of the number of people you like each day?

Accepted Answer

Approximately Normal

Question 7

If you only look through 10 profiles per day with a $$0.1$$ probability of liking each, what is the shape of the distribution of the number of people you like each day?

Accepted Answer

Far from Normal, right-skewed

Question 8

For a sample proportion $$\hat{p}$$ based on $$n$$ independent trials with probability $$p of $$success, what is the mean and standard deviation of $$\hat{p}$$?

Accepted Answer

Mean $$= p$$, Standard deviation $$= \sqrt{\frac{p(1-p)}{n}}$$

Question 9

An insurance company claims that 75% of policyholders have smoke detectors. In a random sample of 80 applications, what is the standard deviation of the sample proportion of clients with smoke detectors?

Accepted Answer

$$\sqrt{\frac{0.75 	imes 0.25}{80}}$$

Question 10

Which of the following best describes the Central Limit Theorem (CLT)?

Accepted Answer

As sample size increases, the sampling distribution of the sample mean approaches a Normal distribution, regardless of the original population distribution.

Question 11

If the original population is highly skewed, what must be true for the sampling distribution of the sample mean to be approximately Normal?

Accepted Answer

The sample size $$n$$ must be large.

Question 12

Suppose human gestation times are Normally distributed with mean $$266$$ days and standard deviation $$16$$ days. What is the standard deviation of the sample mean for a sample of $$100$$ women?

Accepted Answer

$$\frac{16}{\sqrt{100}}$$

Question 13

Which of the following statements is TRUE regarding the sampling distribution of the sample mean?

Accepted Answer

If the original data are Normal, the sampling distribution of the mean is always Normal, regardless of sample size.

Question 14

A company receives 80 applications, and 75% of policyholders are expected to have smoke detectors. What is the expected number of clients with smoke detectors?

Accepted Answer

$$80 	imes 0.75 = 60$$

Question 15

Which of the following is a correct application of the Central Limit Theorem?

Accepted Answer

The average of a large sample of i.i.d. random variables will be approximately Normally distributed.

Question 16

If $$X \sim 	ext{Binomial}(n, p)$$ and $$n is $$large, what is the approximate probability that $$X is $$less than or equal to $$k$$?

Accepted Answer

Use the Normal approximation: $$P(X \leq k) \approx P\left(Z \leq \frac{k+0.5-np}{\sqrt{np(1-p)}}\right)$$

Question 17

Which of the following is NOT a property of a sampling distribution?

Accepted Answer

It is always identical to the population distribution.

Question 18

If the sample size is small and the probability of success is low, what is the shape of the Binomial distribution likely to be?

Accepted Answer

Right-skewed

Question 19

Suppose you want to use the Normal approximation for the sample proportion $$\hat{p}. $$Which condition must be satisfied?

Accepted Answer

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

Question 20

A researcher samples 1000 people and finds the sample proportion of a certain trait. If $$p=0.3$$, what is the standard deviation of the sample proportion?

Accepted Answer

$$\sqrt{\frac{0.3 	imes 0.7}{1000}}$$

Sampling Distribution Models, Normal Approximation, and the Central Limit Theorem

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