Statistics: Single-Sample t Tests, Formulas, and Critical Values

Estimating Parameters

Estimating a population value using information from a sample.

Population Parameter

A characteristic/value of a population, such as the population mean (μ) or population standard deviation (σ).

μ (mu)

The population mean.

σ (sigma)

The population standard deviation.

M

The sample mean.

s

The sample standard deviation; it is used to estimate the population standard deviation σ when σ is unknown.

Sample Mean Formula

M = ΣX / N

N

The number of scores/observations in the sample.

Population Standard Deviation

An exact measure of the spread of scores in a population, represented by σ.

Sample Standard Deviation

The estimate of the population standard deviation, represented by s.

Why do we use N − 1 for the sample standard deviation?

The sample standard deviation typically underestimates the population standard deviation because samples usually have less spread than the population.

Sample Standard Deviation Formula

s = √[Σ(X − M)² / (N − 1)]

X

An individual score/value in the sample.

X − M

The deviation of an individual score from the sample mean.

Σ(X − M)²

The sum of all squared deviations from the sample mean.

Degrees of Freedom (df)

The amount of information that is free to vary in a statistic.

General Degrees of Freedom Formula

df = N − number of parameters estimated in calculating the statistic.

Degrees of Freedom for s

df = N − 1

Why is df = N − 1 for s?

Because the sample mean is estimated when calculating s, which effectively locks one score/value.

What happens as degrees of freedom increase?

The sampling distribution increasingly approximates the ideal distribution; for the mean, it increasingly approximates the z distribution.

Z Distribution

The ideal/reference distribution that the t distribution increasingly resembles as degrees of freedom become large.

Why does the t test use a different distribution than the z test?

The t statistic is an approximation of the ideal z statistic because the population standard deviation σ is unknown and must be estimated with s.

Single Sample t Test

A test used to determine whether a sample is significantly different from a population when the population standard deviation σ is unknown.

When is a single sample t test used?

When you want to compare a sample mean to a known population mean but do not know the population standard deviation.

Single Sample t Test Formula

t(df) = (M − μ) / σM, where df = N − 1.

Estimated Standard Error of the Mean

σM = s / √N

What replaces σ in the t test?

The sample standard deviation s is used to estimate the unknown population standard deviation σ.

Standard Error of the Mean Formula

σM = s / √N

What does the t statistic tell you?

It tells you how far the sample mean is from the population mean in standard-error units.

Numerator of the t formula

M − μ

Denominator of the t formula

σM, the estimated standard error of the mean.

t Test Degrees of Freedom

df = N − 1

How do you report a t statistic?

Report the calculated t value together with its degrees of freedom, such as t(7) = −0.33.

t Distribution

The sampling distribution used for t statistics when the population standard deviation is unknown.

Shape of the t Distribution

The shape depends on the degrees of freedom.

Small df and the t Distribution

At small degrees of freedom, the t distribution has thicker tails and a shorter peak.

Platykurtic

A distribution with relatively thick tails and a shorter peak; the slides describe the t distribution at small df as platykurtic.

What happens to the t distribution as df increases?

It becomes more like the z distribution.

Mesokurtic

The shape the t distribution increasingly approaches as degrees of freedom become larger; around df = 30, the t distribution is basically the same as the z distribution according to the slides.

Critical Value

The cutoff value from the appropriate t distribution that is used to determine whether a calculated t statistic is sufficiently extreme to reject the null hypothesis.

t Table

A table used to find critical values of t based on degrees of freedom and alpha.

Alpha (α)

The significance level used to determine the critical value.

One-Tailed Test

A hypothesis test in which the rejection region is in one direction/tail of the distribution.

Two-Tailed Test

A hypothesis test in which the rejection regions are in both tails of the distribution.

When should you use a two-tailed test?

When the hypothesis predicts that the sample is different from the population without specifying a direction.

When should you use a one-tailed test?

When the hypothesis predicts a specific direction of difference.

'Different' vs. 'Higher/Lower'

A hypothesis saying a sample is 'different' is typically two-tailed; a hypothesis predicting that it is specifically higher or lower is directional and uses one tail.

How do you find a critical t value?

First determine the degrees of freedom, then determine whether the test is one-tailed or two-tailed, then find the appropriate alpha column, and finally read the critical t value from the row matching your df.

Step 1 for finding a critical t value

Calculate df = N − 1.

Step 2 for finding a critical t value

Determine whether the hypothesis is one-tailed or two-tailed.

Step 3 for finding a critical t value

Identify the alpha level, such as α = .05.

Step 4 for finding a critical t value

Find the row in the t table that corresponds to your degrees of freedom.

Step 5 for finding a critical t value

Move across to the correct one-tailed or two-tailed alpha column and read the critical t value.

Critical t: df = 7, one-tailed α = .05

1.895

Critical t: df = 7, two-tailed α = .05

2.365

Critical t: df = 7, one-tailed α = .01

2.998

Critical t: df = 7, two-tailed α = .01

3.500

Critical t: df = 7, one-tailed α = .10

1.415

Critical t: df = 7, two-tailed α = .10

1.895

Critical t: df = 6, one-tailed α = .05

1.943

Critical t: df = 6, two-tailed α = .05

2.447

Critical t: df = 8, one-tailed α = .05

1.860

Critical t: df = 8, two-tailed α = .05

2.306

Critical t: df = 9, one-tailed α = .05

1.833

Critical t: df = 9, two-tailed α = .05

2.262

Important t-table rule

The correct critical value depends on BOTH the degrees of freedom and whether the test is one-tailed or two-tailed, as well as the alpha level.

If df = 7 and the test is two-tailed with α = .05, what critical value do you use?

±2.365. The positive table value is 2.365; for a two-tailed test, the rejection regions are in both tails.

If df = 7 and the test is one-tailed with α = .05, what critical value do you use?

1.895 for the positive/right-tailed test or −1.895 for the negative/left-tailed test.

Critical Value (One-Tailed Test)

1.895 for the positive/right-tailed test or −1.895 for the negative/left-tailed test.

Critical Value (Two-Tailed Test, df=6)

±2.447.

Critical Value (Two-Tailed Test, df=8, α=0.01)

3.355.

Hypothesis Testing with a t Test

The steps are the same as with a z test; report df with t and use the t table rather than the z table.

Null Hypothesis (H₀)

The hypothesis that represents the population value or no predicted difference being tested against.

Alternative Hypothesis (H₁ or Hₐ)

The hypothesis representing the predicted difference between the sample and population.

Decision Rule

Compare the calculated t statistic to the appropriate critical t value to determine whether to reject H₀.

Reject H₀

If the calculated t statistic falls in the rejection region beyond the appropriate critical value.

Fail to Reject H₀

If the calculated t statistic does not fall in the rejection region.

Two-Tailed Rejection Rule

Reject H₀ when |t| is greater than the positive critical t value.

One-Tailed Rejection Rule

Reject H₀ when the calculated t falls beyond the critical value in the predicted direction.

Degrees of Freedom Formula

df = N − 1.

Standard Error Formula

σM = s / √N.

t Statistic Formula

t = (M − μ) / σM.

School Scores Mean

M = 625/8 = 78.125.

School Scores Standard Deviation

s ≈ 16.243.

School t Statistic

t = (78.125 − 80) / 5.743 ≈ −0.326.

School Critical Value

The critical value is ±2.365 for a two-tailed test at α = .05.

Class Scores Mean

M ≈ 12.857 for scores 8, 15, 20, 18, 7, 10, 12.

Library Scores Mean

M = 326/8 = 40.75.

Complete Single Sample t Test Formula

t = (M − μ) / (s / √N), with df = N − 1.

Common Mistake #1

Using N instead of N − 1 for degrees of freedom.

Common Mistake #2

Using the one-tailed critical value when the hypothesis is two-tailed.

Common Mistake #3

Using the two-tailed critical value when the hypothesis is one-tailed.

Most Important Difference Between z and t Tests

For the single-sample t test, σ is unknown and estimated using s.

Key Formulas to Memorize

M = ΣX/N; s = √[Σ(X − M)²/(N − 1)]; σM = s/√N; t = (M − μ)/σM; df = N − 1.