Unit 13: Tests of Significance - Subjective Questions

Definition: If is a standard normal random variable and is an independent chi-square random variable with degrees of freedom, then

follows Student's -distribution with degrees of freedom.

Principal properties:

• It is continuous, bell-shaped, and symmetric about zero.
• Its mean is for .
• Its variance is for .
• It has heavier tails than the standard normal distribution.
• Its shape depends on the degrees of freedom .
• As , the -distribution approaches the standard normal distribution.
• It is commonly used when the population variance is unknown and is estimated from a small sample.

Standard normal distribution and Student's -distribution differ as follows:

• Parameters: The standard normal distribution has no degrees-of-freedom parameter, whereas a -distribution is determined by its degrees of freedom .
• Shape: Both are symmetric and bell-shaped, but the -distribution has heavier tails and a lower central peak.
• Variance: The standard normal distribution has variance . The -distribution has variance when .
• Use: A -test is generally used when the population standard deviation is known or the sample is large. A -test is used when the population standard deviation is unknown, especially for a small sample.
• Convergence: As increases, the -distribution converges to the standard normal distribution.

The heavier tails of the -distribution account for the additional uncertainty caused by estimating the population standard deviation.

A critical value of is a point on the -distribution that separates the rejection region from the non-rejection region of a hypothesis test.

It depends on:

• the significance level ,
• the degrees of freedom , and
• whether the test is one-tailed or two-tailed.

For a right-tailed test, the critical value satisfies

For a left-tailed test, the critical value is .

For a two-tailed test, is divided equally between the two tails, giving critical values

The null hypothesis is rejected if the calculated statistic falls beyond the relevant critical value. For example, in a two-tailed test, rejection occurs when

Critical values are obtained from a -table or statistical software.

Suppose is a random sample from a normal population with unknown mean and unknown variance . We wish to test

The standardized sample mean would be

but is unknown. It is therefore replaced by the sample standard deviation

For a normal population,

and

with and independent. Hence,

Thus, the required statistic is

with degrees of freedom. For a two-tailed test, reject when .

Step 1: State the hypotheses

Step 2: Calculate the test statistic

Here, , , , and .

The degrees of freedom are

Step 3: Apply the decision rule

For a two-tailed test at , reject if

Since , the statistic is not in the rejection region.

Conclusion: We fail to reject . At the level, there is insufficient evidence that the population mean differs from .

A paired -test is used when observations occur in dependent pairs, such as measurements taken before and after treatment on the same subjects or observations matched by relevant characteristics.

Let the pairwise differences be

Their sample mean and standard deviation are

and

To test , the statistic is

which follows a -distribution with degrees of freedom under . Usually, .

Conditions:

• The observations are meaningfully paired.
• Different pairs are independent.
• The differences are measured quantitatively.
• The population of differences is approximately normal, particularly for small .
• The sample is randomly selected or the pairs are randomly assigned.

The test analyzes the differences rather than treating the two samples as independent.

Consider independent samples of sizes and from normal populations with means and . Assume a common but unknown variance:

The unbiased pooled estimate of is

Since

the estimated standard error is

To test

the statistic is

Under , it follows Student's -distribution with

degrees of freedom. The derivation relies on normality, independence, and equality of population variances.

Step 1: Calculate the pooled variance

Therefore,

Step 2: Calculate the standard error

Step 3: Calculate the statistic

The degrees of freedom are

Step 4: Make the decision

Reject if . Since

we fail to reject .

Conclusion: At the significance level, the evidence is insufficient to conclude that the two population means differ.

Pooled two-sample -test:

• Assumes the two populations have equal variances.
• Uses the pooled variance

• Has degrees of freedom.
• Its statistic is

Welch's -test:

• Does not assume equal population variances.
• Uses

• Uses the Welch-Satterthwaite approximation:

Welch's test is generally preferred when variances or sample sizes differ substantially.

Major applications:

• Testing a single population mean when the variance is unknown.
• Testing the difference between two independent means.
• Testing a mean difference using paired observations.
• Constructing confidence intervals for means and mean differences.
• Testing the significance of regression coefficients.
• Testing the significance of a population correlation coefficient under suitable assumptions.

General assumptions:

• Samples are randomly selected.
• Observations within a sample are independent.
• The measured variable is quantitative.
• For exact small-sample procedures, the underlying population or paired differences are normally distributed.
• The pooled two-sample test additionally requires equal population variances.

The -distribution is particularly important for small samples because estimating introduces extra uncertainty. With larger samples, -procedures are often robust to moderate departures from normality, provided there are no severe outliers or extreme skewness.

If and are independent chi-square random variables with and degrees of freedom, respectively, then

has an -distribution with numerator degrees of freedom and denominator degrees of freedom .

Important properties:

• The -distribution is continuous and defined only for .
• It is generally positively skewed.
• Its shape depends on two degrees-of-freedom parameters, and .
• It becomes less skewed as both degrees of freedom increase.
• Its mean is for .
• If , then

• It is widely used for comparing variances and in analysis of variance.

Let independent random samples be drawn from two normal populations with variances and . Their sample variances are and .

For normal populations,

and

Because the samples are independent, these chi-square variables are independent. Therefore,

has an distribution. Simplification gives

Under , this becomes

The larger sample variance is often placed in the numerator for a two-sided test, so . The numerator and denominator degrees of freedom correspond to the samples whose variances appear in those positions.

An upper-tail critical value is defined by

It depends on the significance level and on both numerator and denominator degrees of freedom.

If , then

Therefore, the lower-tail critical point with probability is the reciprocal of an upper-tail point after interchanging the degrees of freedom:

For a two-tailed variance test at level , the rejection regions are

or

When the larger sample variance is always put in the numerator, only the appropriate upper-tail critical value is normally consulted, with careful assignment of degrees of freedom.

Step 1: State the hypotheses

Step 2: Calculate the statistic

The larger sample variance is , so

The degrees of freedom are

Step 3: Make the decision

With the larger variance in the numerator, compare the statistic with the supplied upper-tail critical value for the two-sided test:

Since

the statistic does not fall in the rejection region.

Conclusion: We fail to reject . At the significance level, there is insufficient evidence that the two population variances are different.

One-way analysis of variance tests

against the alternative that at least one population mean differs.

The total variation is divided into:

• Between-group variation, measuring differences among group means.
• Within-group variation, measuring random variation inside groups.

If there are groups and total observations, then

and

The test statistic is

Under , both mean squares estimate the common population variance, so should be near . If the group means differ substantially, tends to exceed , producing a large -value.

The statistic follows under . Reject when

Assumptions: independent observations, normally distributed errors within groups, and equal population variances.

To test a single population mean, state

When the population standard deviation is known, the statistic is

Under , this statistic follows the standard normal distribution if the population is normal. For a large sample, approximate normality follows from the central limit theorem under suitable conditions.

Decision rules at significance level :

• Right-tailed test: reject if .
• Left-tailed test: reject if .
• Two-tailed test: reject if .

Assumptions:

• The sample is random.
• Observations are independent.
• The population standard deviation is known, or a large-sample approximation is justified.
• The population is normal for an exact small-sample -test.

A corresponding two-sided confidence interval is

Step 1: State the hypotheses

Step 2: Calculate the standard error

Step 3: Calculate the test statistic

Step 4: Make the decision

For a two-tailed test at , reject if

Since , reject .

Conclusion: At the significance level, there is significant evidence that the two population means differ.

-based tests:

• Use a statistic that follows or approximately follows .
• Commonly test means and proportions.
• For mean tests, they are exact when is known and normality holds, or approximate for large samples.

-based tests:

• Use statistics involving an estimated standard error.
• Primarily test one mean, paired mean differences, two independent means, and regression coefficients.
• Depend on one degrees-of-freedom parameter.
• Are especially suitable when is unknown.

-based tests:

• Usually involve a ratio of two independent variance estimates.
• Are used to compare two variances, test several means through ANOVA, and test regression models.
• Depend on numerator and denominator degrees of freedom.
• Take only nonnegative values and are generally right-skewed.

A useful relationship is that if , then

Thus, a two-sided -test and the corresponding one-degree-of-freedom -test produce equivalent significance conclusions.

The sampling distribution of the correlation coefficient is not normally distributed, especially when the population correlation is far from zero. Fisher's transformation is

For a bivariate normal population and a sufficiently large sample,

Thus, its standard error is approximately

To test , transform both and and use

where

A confidence interval is first constructed on the transformed scale and then converted back using

The transformation stabilizes the variance and improves normal approximation.

Step 1: Transform the correlations

For the first sample,

For the second sample,

Step 2: Calculate the standard error

For independent correlations,

Step 3: Calculate the statistic

Step 4: Make the decision

For a two-tailed test at , the critical values are . Since

we fail to reject the null hypothesis.

Conclusion: At the level, there is insufficient evidence that the two population correlations differ. The result is very close to the rejection boundary.