Question 1

A researcher is studying the heights of adult women in North America, which are normally distributed with a mean $$\mu = 65$$ inches and a standard deviation $$\sigma = 3.5$$ inches. What is the $$z-$$score for a woman who is 62 inches tall?

Accepted Answer

$$z = \dfrac{62 - 65}{3.5} = -0.86$$

Question 2

Which of the following statements best describes the effect of increasing the sample size from $$n = 100 to$$ $$n = 400 on $$the standard error of the mean, given a population standard deviation $$\sigma = 3.0$$ years?

Accepted Answer

The standard error decreases from $$0.3 to$$ $$0.15$$ years.

Question 3

For a normal distribution, what is the probability that a randomly selected value falls within $$1.44$$ standard deviations of the mean?

Accepted Answer

$$0.90$$

Question 4

A population has a mean $$\mu = 500$$ and standard deviation $$\sigma = 16. If a $$sample of $$n = 100 is $$taken, what is the standard error of the sample mean?

Accepted Answer

$$SE = \dfrac{16}{\sqrt{100}} = 1.6$$

Question 5

Using the Empirical Rule for the Normal Distribution, approximately what percentage of data falls within two standard deviations of the mean?

Accepted Answer

95%

Question 6

A random variable $$X is $$normally distributed with mean $$\mu = 0$$ and standard deviation $$\sigma = 1. $$What is the probability that $$X \u003e 0.84$$?

Accepted Answer

$$0.2005$$

Question 7

Which of the following best describes a sampling distribution?

Accepted Answer

It is the distribution of a statistic (such as the mean) calculated from all possible samples of a given size from a population.

Question 8

If the cumulative probability for $$z = 0.84 is$$ $$0.7995$$, what is the probability that a value is greater than $$z = 0.84$$?

Accepted Answer

$$1 - 0.7995 = 0.2005$$

Question 9

A population distribution is normal with mean $$\mu = 116$$ years and standard deviation $$\sigma = 3.0$$ years. What is the mean and standard deviation of the sample mean for samples of size $$n = 100$$?

Accepted Answer

Mean $$= 116$$, Standard deviation $$= 0.3$$

Question 10

Which of the following is NOT a property of the standard normal distribution?

Accepted Answer

The mean is $$1$$

Question 11

A population has a normal distribution with mean $$\mu = 500$$ and standard deviation $$\sigma = 16. $$According to the Empirical Rule, what is the range within which 99% of sample means will fall for samples of size $$n = 100$$?

Accepted Answer

Between $$468$$ and $$532$$

Question 12

Which of the following best describes the Central Limit Theorem?

Accepted Answer

The sampling distribution of the sample mean approaches a normal distribution as the sample size increases, regardless of the population's distribution.

Question 13

A discrete random variable is defined as:

Accepted Answer

A variable that can take on only specific, countable values.

Question 14

If you want to decrease the standard error of the mean by half, by what factor must you increase the sample size?

Accepted Answer

By a factor of $$4$$

Question 15

Which of the following is a correct formula for the $$z-$$score?

Accepted Answer

$$z = \dfrac{x - \mu}{\sigma}$$

Question 16

A population proportion is defined as:

Accepted Answer

The fraction of the population that possesses a certain characteristic.

Question 17

Which of the following is a property of a probability distribution?

Accepted Answer

The sum of all probabilities is $$1$$

Question 18

A weighted average is most appropriately used when:

Accepted Answer

Different values contribute unequally to the mean due to varying frequencies or importance.

Question 19

Which of the following best describes cumulative probability?

Accepted Answer

The probability that a random variable is less than or equal to a specific value.

Question 20

If a population has a mean $$\mu$$ and standard deviation $$\sigma$$, what is the mean and standard deviation of the sampling distribution of the sample mean for samples of size $$n$$?

Accepted Answer

Mean $$= \mu$$, Standard deviation $$= \dfrac{\sigma}{\sqrt{n}}$$

Statistics Study Guide: Normal Distribution, Sampling, and Standard Error

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