
Single-Sample t test
 Statistical test that compares a sample mean to a
population mean when the population standard
deviation (σ) is not known.
▪ If population standard deviation (σ) is given, do a z-test.
▪ If sample standard deviation (s) is given (not population
standard deviation) or can be calculated, do a t-test.

Example: Dr. Farshad’s reaction time study
 Do adults with attention deficit hyperactivity
disorder (ADHD) differ in reaction time from the
general population?
 Reaction time test for adults in the US,
μ = 200 milliseconds (msec)
 Obtained a random sample of 141 adults diagnosed
with ADHD from ADHD treatment centers in Illinois
 Reaction time for the sample (M = 220, s = 27)

Example: Number of minutes of Netflix viewing per day
 Do people in PSY 305 differ in number of minutes of Netflix
viewing per day compared to the general population?
▪ Netflix released company data that suggested the average
subscriber views 93 minutes of content each day.
▪ Let’s figure out how the mean number of minutes for our
sample (the class) compares to the population mean of 93
minutes.

STEP 1 - Pick a Test

STEP 2 - Check the Assumptions

STEP 3 - List the Hypotheses

STEP 4 - Set the Decision Rule

STEP 5 - Calculate the Test Statistic

STEP 6 - Interpret the Results

STEP 1 - Pick a Test
 We’re comparing mean of a sample to the
mean of a population.
 We do not know population standard deviation,
so we must choose the single-sample t test.

STEP 2 – Check the Assumptions
Assumption
Explanation
Robustness
Random sample
The sample is a random
sample from the
population.
Robust if violated.
Independence of
observations
Cases within the sample
don’t influence each other.
Not robust to violations
Normality
The dependent variable is
normally distributed in the
population.
Robust to violations if
the sample size is
large.
Assumptions for the Single-Sample t Test

STEP 3 – List the Hypotheses
▪ What do we think will be true?
▪ Are we thinking our mean will vary in some way
(either more or less) from the general population in
minutes of Netflix watched per day?
 Two-tailed test
▪ Are we thinking we might be watching more Netflix than
the general population? Or that we might be watching
less Netflix than the general population?
 Either of these would call for a one-tailed test

STEP 3 – List the Hypotheses
 Two-tailed, non-directional, single-sample t test
▪ Don’t know whether we might watch more or less Netflix than
the general population
▪ H0: μStats Class = 93
▪ H1: μStats Class ≠ 93
 One-tailed, directional, single-sample t test
▪ We believe we watch more Netflix than the general population.
▪ H0: μStats Class ≤ 93
▪ H1: μStats Class > 93
 One-tailed, directional, single-sample t test
▪ We believe we watch less Netflix than the general population.
▪ H0: μStats Class ≥ 93
▪ H1: μStats Class < 93

STEP 4 – Set the Decision Rule
 Critical Value of t
▪ Value of t used to determine whether null hypothesis is rejected
or not
▪ Abbreviated tcv
 Three pieces of information are needed to find the tcv
1) Is the test one-tailed or two-tailed?
2) How willing are you to make a Type I error (when the
researcher concludes, mistakenly, that the null hypothesis
should be rejected. False positive. Conclude your results are
significant when they actually occurred by chance)?
-Let’s say only willing to make a Type I error 5% of the time,
α = .05
3) How large is the sample size? N = 30…

Degrees of freedom: the number of values in a
sample that are free to vary when estimating
statistical parameters.
 Ex: Let’s suppose you want to walk a different dog
every day.
N=7

Degrees of freedom: the number of values in a
sample that are free to vary when estimating
statistical parameters.
 Ex: Let’s suppose you want to walk a different dog
every day.
N=7
Degrees of freedom
(df) = N – 1
df = 7-1 = 6
df = 6

Degrees of Freedom (df )
 Number of values in a sample that are free to vary
𝑑𝑓 = 𝑁 − 1
where 𝑑𝑓 = degrees of freedom
𝑁 = sample size
For our study, there are 30 participants in the sample,
so degrees of freedom are calculated like this:
df = 30 − 1 = 29
df = 29

STEP 4 – Set Decision Rule
 Netflix Example
▪ With df = 29, α = .05, two-tailed, border between
1) Two-tailed
the rare and common zones is ± 2.045
2) Alpha level, α = .05 ▪ If observed value of t falls in the rare zone, null
3) N = 30, df = 29
hypothesis is rejected; one fails to reject it if the
observed value falls in the common zone.
Decision rules written mathematically:
Setting the Decision
Rule: Two-Tailed,
Single-Sample t Test IF
the critical value of t is
± 2.045
If t ≤ –2.045 or if t ≥ 2.045, then reject H0
If –2.045 < t < 2.045, then fail to reject H0
t

STEP 5 – Calculate the Test Statistic
𝑀−𝜇
𝑡=
𝑆𝑀
where 𝑡 = 𝑡 value
𝑀 = sample mean
𝜇 = population mean or a specified value
𝑆𝑀 = estimated standard error of the mean (Equation from before, is below)
𝑆𝑀 =
𝑠
𝑁

STEP 5 – Calculate
the Test Statistic
𝑆𝑀 =
 Calculate estimated
standard error of the
mean (sM)
 Calculate t value
 N = 30, s = 19.6084,
M = 43.7500, μ = 93
=
𝑠
𝑁
19.6084
30
𝑀−𝜇
𝑡=
𝑆𝑀
43.7500 − 93
=
3.5800
19.6084
=
5.4772
−49.2500
=
3.5800
=3.5800
= −13.7570
=3.58
= -13.76

STEP 6 – Interpret the Results
 Need to answer three questions
1) Was the null hypothesis rejected?
2) How big is the effect?
3) What is the confidence interval?

STEP 6 – Interpret the Results
1) Was the null hypothesis rejected?
Well… which of the following statements is true?

If t ≤ –2.045 or if t ≥ 2.045, then reject H0
OR

If –2.045 < t < 2.045, then fail to reject H0
We should also report he results in APA format. APA format provides five pieces
of information: (1) what test was done, (2) the number of cases, (3) the value of
the test statistic, (4) the alpha level used, and (5) whether the null hypothesis
was or wasn’t rejected.
In APA format, the results would be (WE WILL FILL THIS IN DURING CLASS)
t(29) = -13.76, p < .05
• The initial t says that the statistical test was a t test.
• The sample size, 30, is present in disguised form. The number 29, in
parentheses, is the degrees of freedom for the t test. For a singlesample t test, df = N – 1. That means N = df + 1. So, if df = 29, N = 29 + 1 =
30.
• The observed t value, -13.76, is reported. This number is the value
of t calculated, not the critical value of t found in the critical value of t table.
Note that APA format requires the value to be reported to two decimal
places, no more and no fewer.
• The .05 tells that alpha was set at .05 (if it is).
• The final part, p < .05, reveals that the null hypothesis was rejected. It
means that the observed result (our calculated t statistic) is a rare result—it
has a probability of less than .05 of occurring when the null hypothesis is
true.
If we stopped after answering only the first interpretation question, this is what
we would write for an interpretation:
There is a statistically significant difference between the mean number of
minutes spent watching Netflix daily for the students in our statistics class and
the number of minutes of Netflix watched in the general United States
population, t(29) = -13.76, p < .05. The students in our class (M = 43.75) watch
fewer minutes of Netflix than does the general public (μ = 93).

STEP 6 – Interpret the Results
 Effect Size (d)
▪ Measure of the degree of impact of the independent variable
on the dependent variable. It’s kind of like a z score—it is a
standard score that allows different effects (measured by
different variables in different studies) to be expressed—and
compared—with a common unit of measurement.
𝑀−𝜇
𝑑=
𝑠
where 𝑑 = the effect size
𝑀 = sample mean
𝜇 = hypothesized sample mean (population mean)
𝑠 = sample standard deviation

STEP 6 – Interpret the Results
 Netflix study
 M = 43.7500, μ = 93, s = 19.6084
 d = -2.51
𝑀−𝜇
𝑑=
𝑠
43.7500 − 93
=
19.6084
−49.2500
=
19.6084
=-2.5117
= -2.51
Effect Sizes in the Social and Behavioral Sciences
“The effect of being in this class falls in the large range,
suggesting that being in this class is associated with
significantly less daily Netflix viewing.”




Also called coefficient of determination (r squared. r2)
Also tells how much impact the IV has on the DV
But is a percentage, ranging from 0% - 100%
Percentage of variability in the outcome (DV) scores that is
accounted for (or predicted) by the explanatory variable (IV).
2
𝑡
𝑟2 = 2
× 100
𝑡 + 𝑑𝑓
where 𝑟 2 = the percentage of variability in the
outcome variable DV that is accounted for
by the explanatory variable (IV)
𝑡 2 = the squared value of 𝑡 (calculated in previous step, but squared)
𝑑𝑓 = the degrees of fredom for the 𝑡 value



The closer r2 is to 100%, the
stronger the effect of the
explanatory variable (IV) is and
the less variability in the outcome
variable (DV) remains to be
explained by other variables.
The closer r2 is to 0%, the
weaker the effect of the
explanatory variable (IV) and the
more variability in the DV exists
to be explained by other
variables.
For the Netflix data, these
calculations would lead to the
conclusion that
r2 = 86.72%
𝑟2
𝑡2
= 2
× 100
𝑡 + 𝑑𝑓
−13.762
=
× 100
−13.762 + 29
189.3376
=
× 100
189.3376 + 29
189.3376
=
× 100
218.3376
= .8672 × 100
= 86.72%
“Being in this class has a large effect
on one’s amount of time watching
Netflix daily. In the present study,
knowing whether someone is in this
class explains 86.72% of the
variability in daily Netflix viewing
minutes.”
Cohen (1988) standards
for r2:
• A small effect is
an r2 ≈ 1%.
• A medium effect is
an r2 ≈ 9%.
• A large effect is
an r2 ≈ 25%.
𝟗𝟓%𝑪𝑰𝝁𝑫𝒊𝒇𝒇 = 𝑴 − 𝝁 ± 𝒕𝒄𝒗 × 𝑺𝑴
𝑤ℎ𝑒𝑟𝑒 95%𝐶𝐼𝜇𝐷𝑖𝑓𝑓 = 𝑡ℎ𝑒 95% 𝑐𝑜𝑛𝑓𝑖𝑑𝑒𝑛𝑐𝑒 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙 𝑓𝑜𝑟 𝑡ℎ𝑒
𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑡𝑤𝑜 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑚𝑒𝑎𝑛𝑠
𝑀 = 𝑡ℎ𝑒 𝑠𝑎𝑚𝑝𝑙𝑒 𝑚𝑒𝑎𝑛 𝑓𝑟𝑜𝑚 𝑜𝑛𝑒 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛
𝜇 = 𝑚𝑒𝑎𝑛 𝑓𝑜𝑟 𝑡ℎ𝑒 𝑜𝑡ℎ𝑒𝑟 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛
𝑡𝑐𝑣 = 𝑡ℎ𝑒 𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑡, 𝒕𝒘𝒐 − 𝒕𝒂𝒊𝒍𝒆𝒅, 𝛼 = .05, 𝑑𝑓 = 𝑁 − 1
𝑆𝑀 = 𝑡ℎ𝑒 𝑒𝑠𝑡𝑖𝑚𝑎𝑡𝑒𝑑 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑒𝑟𝑟𝑜𝑟 𝑜𝑓 𝑡ℎ𝑒 𝑚𝑒𝑎𝑛 (𝑜𝑙𝑑 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛)

STEP 6 – Interpret the Results (calculate 95% confidence
interval for difference between 2 population means)
 μ = 93, M = **, sM = **, tcv = 2.045
 𝟗𝟓%𝑪𝑰𝝁𝑫𝒊𝒇𝒇 = 𝑴 − 𝝁 ± 𝒕𝒄𝒗 × 𝑺𝑴
 𝟗𝟓%𝑪𝑰𝝁𝑫𝒊𝒇𝒇 =
𝟒𝟑. 𝟕𝟓 − 𝟗𝟑 ± 𝟐. 𝟎𝟒𝟓 × 𝟑. 𝟓𝟖𝟎𝟎
 𝟗𝟓%𝑪𝑰𝝁𝑫𝒊𝒇𝒇 =
−𝟒𝟗. 𝟐𝟓
±
𝟕. 𝟑𝟐𝟏𝟏
 From -56.5711 to -41.9289
 From
-56.57 to -41.93
The 95% confidence interval for the difference between population
means ranges from -56.57 to -41.93. In APA format, this confidence
interval would be reported as 95% CI [-56.57, -41.93].

Imagine a race: a person representing the average American
crosses the finish line first, followed by a person representing the
average adult in a stats class.

How much slower (how much less Netflix viewing) is the average
adult in a stats class?

Confidence interval says the average adult in a stats class probably
(there’s a 95% chance) trails the average American in Netflix
viewing by anywhere from 41.93 to 56.57 minutes.

Four points addressed in our interpretation
 Brief explanation of the study
 Present some facts, means of the sample and the
population. Be selective and only report what is
most relevant
 Explain the meaning of the results
 Offer some suggestions for future research,
“replicate”

Replicate
 To repeat a study, usually introducing some
change in procedure to make it better
Here is our full interpretation:
A study compared the amount of daily Netflix viewing minutes of a
sample of statistics students (M = 43.75) to the known amount of Netflix
viewing minutes for the American population (μ = 93). The amount of
daily Netflix viewing for statistics students was statistically significantly
less than the number of minutes found in the general population [t(29) =
-13.76, p < .05]. The size of the difference from the larger population
probably ranges from a 41.93- to a 56.57-minute reduction in Netflix
viewing. This is not a small difference—these results suggest that being
in a statistics class is associated with a large level of reduction in daily
Netflix viewing. If one were to replicate this study, it would be advisable
to obtain a broader sample of adults in statistics classes, not just limiting
it to one class at one university. This would increase the generalizability
of the results.