11.3: Mean Differences, Difference in Means

Question 1

When is a pooled variance used for t-test? What are the conditions? How is it computed?

Answer 1

A pooled variance is used with the t-test for testing the hypothesis that the means of two normally distributed populations are equal and independent, when the variances of the populations are unknown but assumed to be equal.

Null hypothesis testing is the difference between the two means such that it equals to zero. The alternative is that the difference between the two means does not equal to zero.

Can also be tested for variances that are unknown but unequal.

**see equations

Question 2

When is a “paired comparisons” test used? What are the conditions? What is the null hypothesis for two-tailed tests?

Answer 2

Paired comparisons test is used when the observations in the samples are dependent and normally distributed. it is a test of the significance of the mean of the differences between paired observations.

Null hypothesis is testing if the mean of the differences in the pairs is zero such that null hypothesis of mean 1 = mean 2 for two-tailed tests. Alternative is that the mean of the population of paired differences does not equal to zero.

Mean 1 = mean of the population of paired differences
Mean 2 = hypothesized mean of the paired differences, which is commonly zero

Question 3

When is the chi-square test used? Describe the shape of the chi-square distribution. How is it computed?

Answer 3

Chi-square is used for hypothesis tests concerning the variance of a normally distributed population.

Characteristics:

1) The chi-square distribution is asymmetrical
2) Approaches the normal distribution in shape as the degrees of freedom increase
3) Bounded below by zero; cannot be negative

Note that the chi-square values correspond to the probability in the right tail such that 0.025 = probability in the right tail and 0.025 + 0.95 = 0.975 in the left tail.

Null hypothesis for two-tailed tests is testing if the hypothesized variance is true. Alternative is if it is not true.

It is computed with n - 1 degrees of freedom and:

degrees of freedom x sample variance / hypothesized value for the population variance

Question 4

When is F-distributed test statistic used? What are the assumptions? How is it computed?

Answer 4

The hypothesis concerned with the equality of the variances of two populations are tested with an F-distributed test statistic.

Assumptions:

1. Samples are drawn from normal distributions
2. Samples are independent
3. Right-skewed, bounded by 0 on the left
4. When the sample variances are equal, the value of the test statistic is 1 such that the right critical value is always greater than 1 and left critical value is always less than 1

Computed as: F = variance of n1/variance of n2
where the larger variance is in the numerator with n-1 degrees of freedom

Question 5

What is the hypothesis test to address whether the population correlation coefficient is equal to zero?

Answer 5

** see equation

Follows t-distribution with n - 2 degrees of freedom.

Question 6

What is a parametric test? Provide an example.

Answer 6

A parametric test relies on assumptions regarding the distribution of the population and are specific to population parameters.

Example: z-test relies upon a mean and a standard deviation to define the normal distribution. z-test also requires that the sample is large, relying on the central limit theorem to assure a normal sampling distribution.

Question 7

What is a nonparametric test? When is it used (3)?

Answer 7

Nonparametric test do not consider a particular population parameter or have few assumptions about the population that is sampled.

Used when:

1. The assumptions about the distribution of the random variable that support a parametric test are not met. For instance, it is used when the hypothesized value for a variable comes from a non-normal distribution and small sample size so neither the t-test nor z-test is appropriate.
2. The hypothesis does not involve the parameters of the distribution, such as testing whether a variable is normally distributed. Hence, a runs test is used to determine whether the data is random. A runs test provides an estimate of the probability that a series of changes are random.
3. The data are ranks (ordinal) rather than values.

Question 8

What is the Spearman rank correlation test?

Answer 8

The Spearman rank correlation test can be used when the data are not normally distributed.