Q1-R, 2/13

Question 1

What is the primary reason for using a t-test instead of a z-test?

Answer 1

For n<30 we can’t use the z-distribution and must use t-test instead

The t-test is preferred for smaller sample sizes due to the increased variability in estimating the population standard deviation.

Question 2

What does the T-distribution represent?

Answer 2

A family of curves based on degrees of freedom (n-1)

The T-distribution is similar to the normal distribution but has heavier tails, which account for the additional uncertainty in smaller samples.

Question 3

What is the relationship between the t-distribution and the normal distribution as sample size increases?

Answer 3

Approaches normal as n increases

As the sample size increases, the t-distribution becomes more similar to the normal distribution.

Question 4

What are the assumptions when using z or t distributions?

Answer 4

• Data come from a random sample of the population
• The sample average is approximately normal
• Either the data are normal or the CLT applies

The Central Limit Theorem (CLT) states that the distribution of the sample mean approaches a normal distribution as the sample size becomes large.

Question 5

What do p-values indicate?

Answer 5

The probability that our data would have occurred under the null hypothesis

A small p-value suggests that the observed data is unlikely under the null hypothesis, leading to its potential rejection.

Question 6

What does a small p-value imply?

Answer 6

The occurrence is unlikely under the null hypothesis

It does not imply that the null hypothesis is false, just that the data observed is rare under that hypothesis.

Question 7

True or False: The p-value measures the probability that the studied hypothesis is true.

Answer 7

False

The p-value does not provide any information about the truth of the hypothesis itself.

Question 8

True or False: The p-value measures the size of the effect.

Answer 8

False

The p-value does not provide any information about the size of the observed effect.

Question 9

True or False: The p-value tells us the importance of a result.

Answer 9

False

The p-value does not provide any information about the importance of the result.

Question 10

What is the significance level in hypothesis testing?

Answer 10

Our preselected criterion to accept or reject the null hypothesis

Common significance levels include 0.05, 0.01, and 0.10.

Question 11

What differentiates independent samples from dependent samples?

Answer 11

• Independent samples: values in one sample reveal no information about the other
• Dependent samples: values in one sample affect the values in the other

Dependent samples are often paired or matched in some way, such as before-and-after measurements.

Question 12

When to use a z-test in a two-sample test?

Answer 12

• If σ is known, use z-test and use σ instead of s
• If σ is not known but n>30, use z-test

The z-test is appropriate for larger samples where the population standard deviation is known or can be sufficiently estimated.

Question 13

What is the formula for the t-test with independent samples (assumes unequal variances)?

Answer 13

See image

This formula is used when the variances of the two groups being compared are not assumed to be equal.

Question 14

What is the test statistic for dependent samples in a two-sample test?

Answer 14

D=X_1-X_2

Here, D represents the difference between paired observations.

Question 15

How is the t statistic calculated for dependent samples?

Answer 15

See image

Question 16

What should be reported in papers about t-tests?

Answer 16

• Means of each group
• Some measure of variability (typically standard deviation)
• Mean difference between groups (and variability in differences)
• Info on the test: type of test performed, T-value, degrees of freedom, p-value, possibly confidence intervals

Reporting these elements provides a clear understanding of the analysis and its results.