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1. COMPUTER-AIDED DATA ANALYSIS BOOK (978-
978-088-568-7)
CHAPTER 11
One Sample t Test
A one-sample t-test tests the difference between a sample mean and a known or
hypothesized value (the test mean). The test uses the standard deviation of the
sample to estimate σ (the population standard deviation). If the difference between
the sample mean and the test mean is large relative to the variability of the sample
mean, then µ is unlikely to be equal to the test mean. The One-Sample t test
procedure, in essence, determines whether the mean of a single variable differs from
a specified constant. It tests the hypothesis that a population has a particular mean
value, e.g. zero.
Ho: µ = 0 Sample mean is equal to a specified constant i.e. There is no
significant difference (Difference not significant)
Hi; µ ≠ 0 Sample mean is not equal to a specified constant (Two tailed)
There is significant difference Hi; µ ≠ 0 is two directional, meaning
that sample mean can be greater or less than a specified constant
Hi: µ < > 0 (Two-tailed)
Ho: µ ≤ ≥ 0 Sample mean is less than or equal to/ greater than or equal to a specified
constant
Hi: µ < > 0 Sample mean is less than / or greater than a specified constant (One tailed)
A one-sample t test compares the mean of a single column of numbers against a
hypothetical mean you entered. SPSS calculates the t ratio from this equation:
= Sampling Error/Standard Error of the Mean
Statistics - For each test variable: mean, standard deviation, and standard error of
the mean. The average difference between each data value and the hypothesized
test value, a t test that tests that this difference is 0, and a confidence interval for
this difference (you can specify the confidence level
A P value is computed from the t ratio and the numbers of degrees of freedom
(which equals sample size minus 1).
11.1. Why use a one-sample t-test
A one-sample t-test can help answer questions such as:
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2.  Does mean weight differ from the hypothetical test value (1.3 kg in this case)
 Is the average IQ score different from a test value of say, 100?
 Is the mean transaction time on target?
 Does customer service meet expectations?
 Does a water parameter meet WHO standard?
 Is property rent inflation significant over a given number of years?
11.2. Checklist for using a one-sample t test
Before accepting the results of any statistical test, first think carefully about whether
you chose an appropriate test. Before accepting results from a one-sample t test, ask
yourself these questions:
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11.3. SPSS Applications
11.3.1. To Obtain a One-Sample t Test:
From the menus choose: Analyze > Compare Means > One-Sample t Test...
Select one or more variables to be tested against the same hypothesized value.
Enter a numeric test value against which each sample mean is compared.
Optionally, you can click Options to control the treatment of missing data and the level of
the confidence interval.
11.3.2. One-Sample t Test Data Considerations
Data - To test the values of a quantitative variable against a hypothesized test value, choose
a quantitative variable and enter a hypothesized test value.
Assumptions. This test assumes that the data are normally distributed; however, this test is
fairly robust to departures from normality.
11.3.3. One-Sample t Test Options
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11.3.4. To Specify Options for One-Sample t Test
From the menus choose: Analyze > Summarize > One-Sample t Test. In the One-Sample t
Test dialog box, click Options.
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3. 11.4 SPSS DEMO.
The data we will use is given in the table below, with the numbers indicating total
protein (µg/ml).
For our data, double click on the var at the top of the first column or click on the
Variable View tab at the bottom of the page, type in “protein” in the Name column,
and hit Enter.
Under the assumption that you are going to enter numerical data, the rest of the
row is filled in.
Changes in the type and display of the variable can be made by clicking in the
appropriate cells and using any buttons given. Then hit the Data View tab and type in
the data values, following each by Enter.
Save the file as usual where you wish under the name protein.sav. You just need
type protein. The suffix is attached automatically.
We wish to test whether the mean of the population from which the sample came is
70 as opposed to a true mean greater than 70. We test:
Ho : µ = 70
Ha : µ > 70.
From the menu, choose Analyze > Compare Means > One-Sample t Test. Select
protein from the left-hand window and click the right arrow to move it to the Test
Variable(s) window. Set the Test Value to 70.
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4. Click on Options. Set the Confidence Interval to 95% (or any other value you desire.
Then click Continue followed by OK. You get the following output.
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5. SPSS gives us the basic descriptives in the first table. In the second table, we are
given that the t-value for our test is 1.110. The p-value (or Sig. (2-tailed) is given as
.272. Thus the p-value for our one-tailed test is one-half of that or .136. Based on
this test statistic, we would not reject the null hypothesis, for instance, for a value of
α = .05. SPSS also gives us the 95% Confidence Interval of the Difference between
our data scores and the hypothesized mean of 70, namely (-2.6714, 9.3298). Adding
the hypothesized value of 70 to both numbers gives us a 95% confidence interval for
the mean of (67.3286, 79.3298).
If you are only interested in the confidence interval from the beginning, you can just
set the Test Value to 0 instead of 70.
11.5. Interpretations
To make a decision, choose the significance level, α (alpha), before the test:
• If P is less than or equal to α, reject Ho.
• If P is greater than α, fail to reject Ho. (Technically, you never accept Ho, you
simply fail to reject it.)
A typical value for α is 0.05, but you can choose higher or lower values depending on
the sensitivity required for the test and the consequences of incorrectly rejecting the
null hypothesis.
P (.136) > α (.05) we can accept the Ho and conclude that the difference is not
significant i.e. the mean is equal to 70
P-value
The t-test results indicate that
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11.5.1 If the P value is small (one-sample t test)
If the P value is small (usually defined to mean less than 0.05), then it is unlikely that
the discrepancy you observed between sample mean and hypothetical mean is due
to a coincidence arising from random sampling. You can reject the idea that the
difference is a coincidence, and conclude instead that the population has a mean
different than the hypothetical value you entered. The difference is statistically
significant. But is the difference scientifically significant? The confidence interval
helps you decide.
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6. The true difference between population mean and hypothetical mean is probably
not the same as the difference observed in this experiment. There is no way to know
the true difference between the population mean and the hypothetical mean. Prism
presents the uncertainty as a 95% confidence interval. You can be 95% sure that this
interval contains the true difference between the overall (population) mean and the
hypothetical value you entered.
11.5.2 If the P value is large (one sample t test)
If the P value is large, the data do not give you any reason to conclude that the
overall mean differs from the hypothetical value you entered. This is not the same as
saying that the true mean equals the hypothetical value. You just don't have
evidence of a difference.
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Further Reading:
Bryman, A.S and Cramer D. (1997). Quantitaive data analysis with SPSS for
Windows: A guide for social scientists. New York: Routeledge
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