Reading 10 Sampling & Estimation

Question 1

Explain sampling error and give the formula.

Answer 1

The error caused by using a sample instead of the entire population to draw conclusions relating to population parameters. It equals the difference between a sample statistic and the corresponding population parameter.

Sample mean − Population mean =
̅
X
− μ

Question 2

Explain the central limit theorem.

Answer 2

The central limit theorem allows us to make accurate statements about the population mean and variance using the sample mean and variance regardless of the distribution of the population, assuming adequate sample size (n ≥ 30).

Question 3

What should be done when the population is normally distributed?

Answer 3

Use the z-statistic when the population variance is known.

Use the t-statistic when the population variance is not known.

Question 4

Give the formula to calculate the standard error of sample mean when the population variance is known.

Answer 4

σ
––
X
=
σ
√
n

σ
––
X
= the standard error of the sample mean

σ = the population standard deviation

n = the sample size

Note: The standard deviation of the distribution of sample means is also known as the standard error of the sample mean.

Question 5

Give the formula used for the t-distribution to construct confidence intervals when the variance of a normally distributed population is not known.

Answer 5

––
X
±
t
α
/
2
s
√
n

––
X
= sample mean (the point estimate of the population mean)

t
α
/
2
 = the t-reliability factor

s
√
n
= standard error of the sample mean

s = sample standard deviation

Question 6

How is the confidence interval calculated for the population mean when the population follows a normal distribution and its variance is known?

Answer 6

––
X
±
Z
α
/
2
σ
√
n

where:

––
X
= The sample mean (point estimate of population mean).

Zα/2= The standard normal random variable for which the probability of an observation lying in either tail is α / 2 (reliability factor).

σ
/
√
n
= The standard error of the sample mean.

Question 7

Describe when data-mining bias occurs.

Answer 7

Researchers have not formed a hypothesis in advance and are therefore open to any hypothesis suggested by the data.

When researchers narrow the data used to reduce the probability of the sample refuting a specific hypothesis.

Question 8

Identify the properties of a student’s bell-shaped probability distribution (t-distribution). Name the 2 scenarios in which a t-distribution is used.

Answer 8

Symmetrical with lower peak and fatter tails than the normal curve.

Defined by degrees of freedom (df), equal to sample size minus one (n − 1).

As the degrees of freedom increase, the shape of the t-distribution approaches the shape of the standard normal curve.

Two scenarios:

Construct confidence intervals for a normally distributed population with unknown variance and small sample size (n < 30).
For a non-normally distributed population with unknown variance and large sample size (n ≥ 30 based on the central limit theorem).

Question 9

Define confidence interval, and list the reliability factors frequently used when constructing confidence intervals based on the standard normal distribution.

Answer 9

A confidence interval uses sample data to calculate a range of values for an unknown population parameter given probability (1 − α), the degree of confidence that the relevant parameter will lie in the computed interval. α is the level of significance.Following are the reliability factors”
For a 90% confidence interval use z0.05 = 1.65
For a 95% confidence interval use z0.025 = 1.96
For a 99% confidence interval use z0.005 = 2.58