One-Sample t-Test

One-Sample t-Test
Example 1: Mortgage Process Time
Problem Data set
A faster loan processing time produces higher productivity Mortgage.MPJ
and greater customer satisfaction. A financial services
institution wants to establish a baseline for their process by Variable Description
estimating their mean processing time. They also want to
determine if their mean time differs from a competitor’s claim Loan Loan application number
of 6 hours. Hours Number of hours until customer receives notification
Data collection
A financial analyst randomly selects 7 loan applications and
manually calculates the time between loan initiation and when
the customer receives the institution’s decision.

Tools
• Descriptive Statistics
• 1-Sample t
• Normality Test
• Time Series Plot
• Individual Value Plot

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Displaying descriptive statistics

Use descriptive statistics to summarize important features of Display Descriptive Statistics
the data. In particular, descriptive statistics provide useful
information about the location and variability of the data. 1. Open Mortgage.MPJ.

2. Choose Stat > Basic Statistics > Display Descriptive

Statistics.

3. In Variables, enter Hours.

4. Click OK.

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Interpreting your results

The statistics indicate that the sample mean is 5.079 hours.
This is slightly below the target time of 6 hours. The one
sample t-test compares this difference (0.921) to the variation
in the data. While the standard deviation of the sample data
is 1.319 hours, the one sample t-test uses the SE Mean.

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Standard Error of the Mean

We can see that the SE Mean is simply the standard deviation
divided by the square root of the number of data points and
represents the dispersion or variation in the distribution of
sample means. The one-sample t-test uses the distribution of
the sample mean (not the distribution of the data) for the
analysis. Therefore, the standard error of the mean will be
used as the estimate of variation for the t-test and confidence
interval.

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Hypothesis testing
What is a hypothesis test Why use a hypothesis test
A hypothesis test uses sample data to test a hypothesis about Hypothesis testing can help answer questions such as:
the population from which the sample was taken. The
one-sample t-test is one of many procedures available for • Are turn-around times meeting or exceeding customer
expectations?
hypothesis testing in Minitab.
For example, to test whether the mean duration of a
• Is the service at one branch better than the service at another?
transaction is equal to the desired target, measure the duration For example,
of a sample of transactions and use its sample mean to
estimate the mean for all transactions. Using information from • On average, is a call center meeting the target time to answer
a sample to make a conclusion about a population is known customer questions?
as statistical inference. • Is the mean billing cycle time shorter at the branch with a new
billing process?
When to use a hypothesis test
Use a hypothesis test to make inferences about one or more
populations when sample data are available.

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One-sample t-test
What is a one-sample t-test Why use a one-sample t-test
A one-sample t-test helps determine whether μ (the A one-sample t-test can help answer questions such as:
population mean) is equal to a hypothesized value (the test
mean). • Is the mean transaction time on target?

The test uses the standard deviation of the sample to estimate

• Does customer service meet expectations?
σ (the population standard deviation). If the difference For example,
between the sample mean and the test mean is large relative
to the variability of the sample mean, then μ is unlikely to be • On average, is a call center meeting the target time to answer
equal to the test mean. customer questions?

When to use a one-sample t-test

• Is the billing cycle time for a new process shorter than the
current cycle time of 20 days?
Use a one-sample t-test when continuous data are available
from a single random sample.
The test assumes the population is normally distributed.
However, it is fairly robust to violations of this assumption for
sample sizes equal to or greater than 30, provided the
observations are collected randomly and the data are
continuous, unimodal, and reasonably symmetric (see [1]).

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 One-Sample t-Test

Testing the null hypothesis

The company wants to determine whether the mean time for 1-Sample t
the approval process is statistically different from the
competitor’s claim of 6 hours. In statistical terms, the process 1. Choose Stat > Basic Statistics > 1-Sample t.
mean is the population mean, or μ (mu).
2. Complete the dialog box as shown below.
Statistical hypotheses
Either μ is equal to 6 hours or it is not. You can state these
alternatives with two hypotheses:
The null hypothesis (H0): μ is equal to 6 hours.

The alternative hypothesis (H1): μ is not equal to 6 hours.

Because the analysts will not measure every loan request in

the population, they will not know the true value of μ.
However, an appropriate hypothesis test can help them make
an informed decision. For these data, the appropriate test is
a 1-sample t-test.

3. Click OK.

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Interpreting your results

The logic of hypothesis testing
All hypothesis tests follow the same steps:
1. Assume H0 is true.

2. Determine how different the sample is from what you

expected under the above assumption.

3. If the sample statistic is sufficiently unlikely under the

assumption that H0 is true, then reject H0 in favor of H1.

For example, the t-test results indicate that the sample mean
is 5.079 hours. The test answers the question, “If μ is equal to
6 hours, how likely is it to obtain a sample mean this different
(or even more different)?” The answer is given as a probability
value (P), which for this test is equal to 0.114.

Test statistic
The t-statistic (-1.85) is calculated as:
t = (sample mean – test mean) / SE Mean

where SE Mean is the standard error of the mean (a measure

of variability). As the absolute value of the t-statistic increases,
the p-value becomes smaller.

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Interpreting your results

Making a decision
To make a decision, choose the significance level, α (alpha),
before the test:

• If P is less than or equal to α, reject H0.

• If P is greater than α, fail to reject H0. (Technically, you
never accept H0. 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. Assuming an α-level of 0.05 for the mortgage
data, not enough evidence is available to reject H0 because
P (0.114) is greater than α.

What’s next
Check the assumption of normality.

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Testing the assumption of normality

The 1-sample t-test assumes the data are sampled from a Normality Test
normally distributed population.
1. Choose Stat > Basic Statistics > Normality Test.
Use a normality test to determine whether the assumption of
normality is valid for the data. 2. Complete the dialog box as shown below.

3. Click OK.

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 One-Sample t-Test

Interpreting your results

Use the normal probability plot to verify that the data do not
deviate substantially from what is expected when sampling
from a normal distribution.
• If the data come from a normal distribution, the points will
roughly follow the fitted line.
• If the data do not come from a normal distribution, the
points will not follow the line.

Anderson-Darling normality test

The hypotheses for the Anderson-Darling normality test are:
H0: Data are from a normally distributed population

H1: Data are not from a normally distributed population

Using an α-level of 0.05, there is insufficient evidence to

suggest the data are not from a normally distributed
population. What’s next
Conclusion Sort the data in time order to check the data for non-random
patterns over time.
Based on the plot and the normality test, assume that the data
are from a normally distributed population.

Note When data are not normally distributed, you may be able to
transform them using a Box-Cox transformation or use a nonparametric
procedure such as the 1-sample sign test.

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Sorting data
You can use Minitab’s Sort command to sort the data in Sort
ascending or descending order—numerically, alphabetically,
or by date. In this example, sort the data by loan application 1. Choose Data > Sort.
number, in ascending, or increasing, order. Sorting the data
by loan application number will put it in time order because 2. Under Column in Level 1, enter Loan.
the loan application numbers are assigned in increasing order 3. In Storage location for the sorted columns, select In the
by the system. Sorting the data chronologically makes it easy
original columns.
to plot the data over time and evaluate it for patterns or
trends. 4. Click OK.
Note Include all appropriate columns in the sorting step, to preserve
the connection between the columns of data.

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Testing the randomness assumption

Use a time series plot to look for trends or patterns in your Time Series Plot
data, which may indicate that your data are not random over
time. 1. Choose Graph > Time Series Plot.

2. Choose Simple, then click OK.

3. Complete the dialog box as shown below.

4. Click OK.

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Interpreting your results

If a trend or pattern exists in the data, we would want to
understand the reasons for them. In this case, the data do not
exhibit obvious trends or patterns.

What’s next
Calculate a confidence interval for the true population mean.

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Confidence intervals
What is a confidence interval Why use a confidence interval
A confidence interval is a range of likely values for a population Confidence intervals can help answer many of the same questions
parameter (such as μ) that is based on sample data. For as hypothesis testing:
example, with a 95% confidence interval for μ, you can be 95%
confident that the interval contains μ. In other words, 95 out • Is μ on target?
of 100 intervals will contain μ upon repeated sampling. • How much error exists in an estimate of μ?

When to use a confidence interval • How low or high might μ be?

Use a confidence interval to make inferences about one or For example,

more populations from sample data, or to quantify the
precision of your estimate of a population parameter, such as • Is the mean transaction time longer than 30 seconds?
μ. • What is the range of likely values for mean daily revenue?

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Using the confidence interval

In the previous analysis, you used a hypothesis test to 1-Sample t
determine whether the mean of the mortgage processing
time was different from the target value. You can also use a 1. Choose Stat > Basic Statistics > 1-Sample t.
confidence interval to evaluate this difference.
2. Click Graphs.
The output pane results for 1-Sample t include values for the
upper and lower bounds of the 95% confidence interval. 3. Complete the dialog box as shown below.
Obtain a graphical representation of the interval by selecting
Individual value plot in the Graphs subdialog box.

4. Click OK in each dialog box.

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Interpreting your results

Confidence interval
The confidence interval is a range of likely values for μ. Minitab
displays the interval graphically as a blue line on the individual
value plot.
Under repeated sampling from the same population, the
confidence intervals from about 95% of the samples would
include μ. Thus, for any one sample, you can be 95% confident
that μ is within the confidence interval.

Note A confidence interval does not represent 95% of the data; this is
a common misconception.

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Interpreting your results

Hypothesis test
The middle tick mark, labeled X, represents the mean of the
sample and the red circle, labeled H0, represents the
hypothesized population mean (6 hours). You can be 95%
confident that the process mean is at least 3.859 hours, and
at most 6.298 hours.
Use the confidence interval to test the null hypothesis:
• If H0 is outside the interval, the p-value for the hypothesis
test will be less than 0.05. You can reject the null hypothesis
at the 0.05 α-level.
• If H0 is inside the interval, the p-value will be greater than
0.05. You cannot reject the null hypothesis at the 0.05
α-level.

Because H0 falls within the confidence interval, you cannot

reject the null hypothesis. Not enough evidence is available
to conclude that μ is different from the target of 6 hours at
the 0.05 significance level.

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Final considerations
Summary and conclusions
According to the t-test and the sample data, you fail to reject
the null hypothesis at the 0.05 α-level. In other words, the
data do not provide sufficient evidence to conclude the mean
processing time is significantly different from 6 hours.
The normality test and the time series plot indicate that the
data meet the t-test’s assumptions of normality and
randomness.
The 95% confidence interval indicates the true value of the
population mean is between 3.859 hours and 6.298 hours.

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Final considerations
Hypotheses Assumptions
A hypothesis test always starts with two opposing hypotheses. Each hypothesis test is based on one or more assumptions about
the data being analyzed. If these assumptions are not met, the
The null hypothesis (H0): conclusions may not be correct.

• Usually states that some property of a population (such The assumptions for a one-sample t-test are:
as the mean) is not different from a specified value or from
a benchmark. • The sample must be random.

• Is assumed to be true until sufficient evidence indicates • Sample data must be continuous.
the contrary. • Sample data should be normally distributed (although this
• Is never proven true; you simply fail to disprove it. assumption is less critical when the sample size is 30 or more).

The alternative hypothesis (H1): The t-test procedure is fairly robust to violations of the normality
assumption, provided that observations are collected randomly
• States that the null hypothesis is wrong. and the data are continuous, unimodal, and reasonably symmetric
(see [1]).
• Can also specify the direction of the difference.
Confidence interval
Significance level
The confidence interval provides a likely range of values for μ (or
Choose the α-level before conducting the test. other population parameters).

• Increasing α increases the chance of detecting a difference, You can conduct a two-tailed hypothesis test (alternative
but it also increases the chance of rejecting H0 when it is hypothesis of ≠) using a confidence interval. For example, if the
actually true (a Type I error). test value is not within a 95% confidence interval, you can reject
H0 at the 0.05 α-level. Likewise, if you construct a 99% confidence
• Decreasing α decreases the chance of making a Type I interval and it does not include the test mean, you can reject H0
error, but also decreases the chance of correctly detecting at the 0.01 α-level.
a difference.

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