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Contents
1 Lesson Outcomes
2 What If We Don't Know σ ?
3 Hypothesis Tests
3.1 Body Temperatures Revisited
3.2 Baby Boom
4 Confidence Intervals
4.1 Automatic Language Translation Programs
4.1.1 Requirements
4.2 Euro Weights
5 Summary
1 Lesson Outcomes
By the end of this lesson, you should be able to:
Compute a confidence interval for a single mean with sigma unknown
Check if the requirements for creating a confidence interval for a single mean with sigma unknown are satisfied
Conduct a hypothesis test for one mean with sigma unknown
Check if the requirements of the hypothesis test for one mean with sigma unknown are satisfied
Interpret a confidence interval for a single mean with sigma unknown in context
Make an informed decision based on the results of a hypothesis test for one mean with sigma unknown
2 What If We Don't Know σ ?
In practice, we almost never know the population standard deviation, σ . So, it is generally not appropriate to use the formula
x̄ − μ
z =
−
σ/√n
In 1908, William Sealy Gosset published a solution to this problem . He found a way to appropriately compute the confidence
interval for the mean when σ is not known. The basic idea is to use the sample standard deviation, s in the place of the true
population standard deviation, σ . If σ is not known, we cannot base the calculations on the standard normal distribution, and we
cannot use the formula above to conduct hypothesis tests.
In a remarkable piece of work, Gosset found the appropriate distribution to use when σ is unknown. At the time of this discovery,
Gosset worked for the Guinness brewery. To avoid problems with industrial espionage, Guinness prohibited employees from
publishing any research results. Knowing his work provided a significant contribution to Statistics, Gosset chose to publish his
results anyway. He chose the pseudonym "Student". Gosset's test statistic was denoted by the letter t , this distribution has come to
be known as Student's t-distribution.
x̄ − μ
t =
−
s/√n
Historical Note:
William Sealy Gosset
(1876-1937) was a
British industrial scientist
and statistician best
known for his discovery
of the t-distribution.
The t -distribution is bell-shaped and symmetrical. The t -distribution has a mean of 0, but it has more area in the tails than the standard normal distribution. The
exact shape of the t -distribution depends on a parameter called the degrees of freedom (abbreviated df ). The degrees of freedom is related to the sample
size. As the sample size goes up, the degrees of freedom increase accordingly. For the procedures discussed in this lesson, the degrees of freedom equal the
sample size minus one: df = n − 1.
Click here (http://www.stat.tamu.edu/~west/applets/tdemo1.html) to explore how the shape of the t -distribution changes with the degrees of freedom. Notice that
as the degrees of freedom increases, the shape of the t -distribution (black curve) gets closer to the standard normal distribution (red curve). The red curve is
identical in the three images below.
t -distribution
t -distribution
with df
with df
= 1
= 5
t -distribution
with df = 15
(Image credit: Webster West, http://www.stat.tamu.edu/~west/applets/tdemo1.html)
3 Hypothesis Tests
3.1 Body Temperatures Revisited
We will apply the t -distribution to the body temperature data we explored previously.
This hypothesis test is conducted in a manner similar to a test for a single mean where σ is known, except that instead of using the population standard deviation,
σ , in the calculations, we estimate this value using the sample standard deviation, s . This leads to a t -distribution, rather than a normal distribution for the test
statistic. We will not need to compute the value of this test statistic by hand. It will be done using Software.
Summarize the relevant background information
We want to conduct a hypothesis test to determine if the mean body temperature is different from 98.6° Fahrenheit. Previously, we assumed that we knew the
value of σ . Actually, this value is not known.
State the null and alternative hypotheses and the level of significance
H o : μ = 98.6
H a : μ ≠ 98.6
α = 0.05
Describe the data collection procedures
We will use the body temperature data, BODYTEMP , collected by Dr. Mackowiak and his colleagues to conduct the test.
Give the relevant summary statistics
x̄ = 98.23
s = 0.738
n = 148
Make an appropriate graph (e.g. a histogram) to illustrate the data
Verify the requirements have been met
We assume that the individuals chosen to participate in the study represent a (simple) random sample from the population.
x̄ will be normally distributed, because the sample size is large. (Note: We could have also noticed that the body temperature data appears to be
normally distributed, so even with a small sample size, x̄ would be normal.)
Give the test statistic and its value
We will need to conduct the analysis using software, so we can report this value.
Instructions for conducting a test for one mean with sigma unknown:
Excel
Here are the instructions for conducting the one sample t-test in
Excel:
The Excel file needed for this analysis is
QUANT IT AT IVEINFERENT IALPROCEDURES.XLS. We will use this
Excel file to conduct the hypothesis tests for a single mean with σ
unknown.
After opening the file, please click on the tab labeled
"One-sample t-test".
Paste the data in the appropriate place in Column A.
Set the null hypothesis value in cell D10. For this
example, the value should be 98.6.
Click in cell D11 and use the drop down menu to set
the alternative hypothesis to: "Not Equal To"
The image below shows the Excel file after these changes have
been made.
SPSS
Here are the instructions for conducting the one sample t-test in SPSS:
Within SPSS, select the menu item:
Analyze → C ompare Means → One − S ample T T est
Move the variable containing your data values into the "Test Variable(s)" box.
Your null hypothesis will state that μ is equal to some number. Enter that
number as the "Test Value." (This is a step that people often forget to do! Don't
forget to do this. This tells SPSS what the value of μ is in your null hypothesis.)
Click on [OK].
The output will contain a box labeled "One-Sample Test." This will give the
value of the test statistic (t ), the degrees of freedom (df ), and "Sig.(2-tailed)."
t
SPSS uses the term "Sig.(2-tailed)" for the area in both tails under the t distribution that is more extreme than the test statistic (t ).
How do we use the "Sig.(2-tailed):" output to find the P -value? This depends on the
alternative hypothesis. Consider each of the following alternative hypotheses that could be
used in a test for a single mean where the null hypothesis is H o : μ = 98.6.
H a : μ ≠ 98.6
The P -value is the area in the two tails beyond the test
statistic, t . In other words, the P -value is equal to "Sig.(2tailed)."
Consider each of the following alternative hypotheses that could
be used in a test for a single mean where the null hypothesis is
H o : μ = 98.6.
H a : μ ≠ 98.6
Choose "Not Equal To" in the
drop-down menu in cell D11.
H a : μ < 98.6
Choose "Less Than" in the dropdown menu in cell D11.
H a : μ > 98.6
Choose "Greater Than" in the
drop-down menu in cell D11.
You will have opportunities to practice using each of these.
H a : μ < 98.6
The P -value is the area in the tail to the left of the test
statistic, t . Use "Sig.(2-tailed)," which gives the area in
both tails, then adjust it so that you get the area to the left.
Usually this means you simply divide "Sig.(2-tailed)" by 2.
The key is to find the area to the left of the test statistic.
H a : μ > 98.6
The P -value is the area in the tail to the right of the test
statistic, t . Use "Sig.(2-tailed)," which gives the area in
both tails, then adjust it so that you get the area to the right.
Usually this means you simply divide "Sig.(2-tailed)" by 2.
The key is to find the area to the right of the test statistic.
If the value of t is large enough, SPSS gives the value of "Sig.(2-tailed)" as ".000". By
double-clicking twice on this cell, SPSS will display more digits. If t is very large, when you
click on this cell, SPSS may show the value "0.000000". Like the normal curve, the t curve
never touches the horizontal axis. So, there is always a little bit of area in the tails, no matter
how far out you go. If SPSS gives a value of 0.000000, report the area as:
< 0.0000005. The seventh digit to the right of the decimal place is unknown, but it must
be less than 5 or else SPSS would report something like 0.000001. So, if the area is given
as 0.000000, it is appropriate to consider it as < 0.0000005. If you need to find a P value for a one-sided test, then this number should be divide by two and reported as
< 0.00000025.
The interpretation of the results will follow the pattern established in the previous hypothesis tests. If the P -value is less than the α level, we reject the null
hypothesis. If the P -value is greater than the α level, we fail to reject the null hypothesis. This is true for every hypothesis test.
The test statistic is t and its value is −6.029. So, we have t
= −6.029.
Notice this is the same number we get if we use the formula:
x̄ − μ
t =
−
s/√n
98.234 − 98.9
≈
−
−
−
0.738/√ 148
≈ −6.03
Any differences are due to rounding.
State the degrees of freedom
In Excel, the degrees of freedom (df ) are given in cell E21. In SPSS, the degrees of freedom (df ) are given in the output in the box labeled "One-Sample
Test".
df = 147
Find the
P
-value and compare it to the level of significance
The P -value is given in the software as "1.2723E-08". Writing this properly and comparing it to α, we have:
P -value = 1.2723 × 10
−8
= 0.000 000 012 723 < 0.05 = α
State your decision
The interpretation of the results will follow the pattern established in the previous hypothesis tests. If the P -value is less than the α level, we reject the null
hypothesis. If the P -value is greater than the α level, we fail to reject the null hypothesis. This is true for every hypothesis test.
Since the P -value was less than α, we reject the null hypothesis.
Present your conclusion in an English sentence, relating the result to the context of the problem
There is sufficient evidence to suggest that the mean body temperature is not 98.6° Fahrenheit.
3.2 Baby Boom
Summarize the relevant background information
The birth weight of a child is an important indicator of their neonatal health. It is important that pediatric health care providers track changes in the birth weights
over time. The birth weight of children in Australia has historically had a population mean of 3373 grams. Is this still the mean birth weight of Australian children, or
has there been a change? We will use the 0.05 level of significance.
State the null and alternative hypotheses and the level of significance
Ho :
μ = 3373
Ha :
μ ≠ 3373
α = 0.05
α = 0.05
Describe the data collection procedures
The birth weights of all children born on December 18, 1997 at the Mater Mothers' Hospital in Brisbane, Australia were recorded . The time of birth (on a 24
hour clock), gender, and birth weight of each child are given in the file BABYBOOM.
Using this data set, test the hypothesis that the mean weight of babies born in Australia is 3373 grams. Use the α
Make an appropriate graph of the data.
= 0.05 level of significance
for this problem.
Answer the following questions:
1. Give the relevant summary statistics
Excel
SPSS
When the data are entered into the file QUANTITATIVEINFERENTIALP ROCEDURES.XLS, When the data are entered into SPSS, the following output is generated:
the following output is generated:
The relevant summary statistics are:
x̄ = 3275.95g
s = 528.03g
n = 44
2. Make an appropriate graph to illustrate the data
Verify the requirements have been met
The data show a left-skewed shape, however the sample size is large. Using the Central Limit Theorem, we can conclude that the sample mean is normally
distributed and the requirements are satisfied.
Answer the following questions:
3. Give the test statistic and its value
The test statistic is a t . The value of the test statistic is −1.219. This was taken from the output above. We have
t = −1.219.
4. State the degrees of freedom
The degrees of freedom are given in the output above:
df = 43
Notice that this is one less than than sample size. For a test for a single mean, the degrees of freedom are always equal to one less than the sample size.
Mark the test statistic and P -value on a graph of the sampling distribution
The P -value is shaded in green:
Find the
P
-value and compare it to the level of significance
P -value = 0.228 > 0.05 = α
State your decision
Since the P -value is greater than the level of significance, we fail to reject the null hypothesis.
Present your conclusion in an English sentence, relating the result to the context of the problem
There is insufficient evidence to suggest that the mean weight of babies born in Australia is different from 3373 grams.
4 Confidence Intervals
The procedure for finding confidence intervals when σ is not known is very similar to the process when σ is known. As a reminder, here is the confidence interval
(from Lesson 10) when σ is known.
(x̄ − z
∗
σ
−
√n
, x̄ + z
∗
σ
−
√n
)
To construct the confidence interval when σ is not known, we replace the population standard deviation, σ , with its point estimate, s. The appropriate
∗
∗
distribution is a t , rather than a z . So, we replace z with t . So the confidence interval when σ is not known is
(x̄ − t
∗
s
−
√n
, x̄ + t
∗
s
−
√n
)
∗
where t is the t-score corresponding to the confidence level and s is the sample standard deviation.
∗
We will use use either Excel or SPSS to construct confidence intervals when σ is unknown. The values of t and s will be computed for us. Examples of this are
given below. You can see the details by clicking on [Show Hand Calculations] in Step 4.
4.1 Automatic Language Translation Programs
Summarize the relevant background information
Computer software is commonly used to translate text from one language to another. As part of his Ph.D. thesis, Philipp Koehn developed a phrase-based
translation program called Pharaoh.
The quality of the translation can vary. A good translation system should match a professional human translation. It is important to be able to quantify how good
the translations produced by Pharaoh are.
The IBM T. J. Watson Research Center developed methods to measure the quality of a translation from one language to another. One of these is the the
BiLingual Evaluation Understudy (BLEU). BLEU is a score ranging from 0 to 1 that indicates how well a computer translation matches a professional human
translation of the same text. Higher scores indicate a better match. BLEU helps companies who develop translation software "to monitor the effect of daily
changes to their systems in order to weed out bad ideas from good ideas".
Describe the data collection procedures
To test Pharaoh's ability to translate, Koehn took a random sample of 100 blocks of Spanish text, each of which contained 300 sentences, and used Pharaoh to
translate each of these to English. The BLEU score was calculated for each of the 100 blocks. The data were extracted from Figure 2 in a paper Koehn
published. The 100 BLEU scores are given in BLEU-SCORES.
Koehn wants to find an estimate of the true mean BLEU score for text translated by the Pharaoh computer program. He would like to compute a confidence
interval, but he does not know the true population standard deviation, σ .
Give the relevant summary statistics
Excel
SPSS
Copying the BLEU-SCORES data into the
QUANT IT AT IVEINFERENT IALPROCEDURES.XLS file in Excel, we get the following:
We can find the summary statistics for the BLEU-SCORES data using the SPSS
command Analyze → Descriptive S tatistics → Explore :
The summary statistics are:
x̄ = 0.2876
s = 0.0264
n = 100
Make an appropriate graph to illustrate the data
Here is a histogram of the data showing 10 bins.
Verify the requirements have been met
The requirements for creating a confidence interval for a mean with σ unknown are the same as the requirements for this procedure when σ is known:
4.1.1 Requirements
There are two requirements that need to be checked when computing a confidence interval for a mean with σ known:
1. A simple random sample was drawn from the population
2. x̄ is normally distributed
Recall the requirement of normality is satisfied if the data are approximately normally distributed or if the sample size is large.
The data are bell-shaped and fairly symmetric. So, the sample mean, x̄ , is approximately normally distributed.
Find the confidence interval
The formula for the confidence interval where σ is known was given in the reading titled Inference for One Mean: Sigma Known (Confidence
Interval)#11:CIformula-SigmaKnown|Lesson 10: Inference for One Mean: Sigma Known (Confidence Interval)|Inference for One Mean: Sigma Known
(Confidence Interval) as
(
−
∗
,
+
∗
)
∗
(x̄ − z
σ
−
√n
, x̄ + z
σ
∗
−
√n
)
It is impossible to know the true standard deviation of the BLEU scores for a new translation program like Pharaoh. Replacing σ with s and replacing z
∗
t , we get:
(x̄ − t
∗
s
−
√n
, x̄ + t
∗
with
s
∗
−
√n
)
∗
The value of t depends on the level of confidence and the sample size. It must be computed using software or looked up on a table. If you are asked to
∗
compute a confidence interval for a mean where the population standard deviation is unknown, the value of t will be given to you. The other numbers (x̄ , s,
and n ) can all be obtained directly from your data.
∗
If we want to create a 95% confidence interval for μ , with 99 degrees of freedom, then t
= 1.9842.
Using the sample statistics:
x̄ = 0.2876
s = 0.0264
n = 100
The 95% confidence interval for μ is:
0.0264
(0.2876 − 1.9842
−
−
− , 0.2876 + 1.9842
√ 100
which reduces to:
(0.2824, 0.2928)
0.0264
−
−
−
√100
)
Excel
SPSS
We will use Excel to find the confidence intervals for the mean.
We can find the confidence intervals for the BLEU-SCORES data using the SPSS
command Analyze → Descriptive S tatistics → Explore :
Do the following:
Open the file QUANT IT AT IVEINFERENT IALPROCEDURES.XLS
Click on the tab labeled "One-sample t-test"
Enter the data, and
Set the confidence level.
In this case, we are using a 95% confidence level. So, you set the confidence
level in the Excel file to 0.95 (i.e., 95%). The confidence interval is given to you
in cells G21 and H21.
By default a 95% confidence interval is shown. To find other confidence
intervals, go to
Analyze → Descriptive S tatistics → Explore . Move the
desired variable into the "Dependent List" and then press the "Statistics" button
on the right of the window. Change the percentage next to "Confidence Interval
for Mean" to the desired confidence level then press "Continue."
A quick word of caution on something that many students get confused on.
Using the Explore function is the best way to calculate confidence intervals.
You may notice that the output for doing a hypothesis test contains a category
labeled "95% Confidence Interval of the Difference." This is different from the
Confidence Interval of the Mean given in the Explore function and will give
you different values.
The 95% confidence interval for the true mean BLEU score is:
(0.282, 0.293)
Present your observations in an English sentence, relating the result to the context of the problem
We are 95% confident that the true mean BLEU score for all translations by the Pharaoh program is between 0.2824 and 0.2928.
Answer the following questions using the data set BLEU-SCORES, which gives the BLEU scores for n
computer program Pharaoh.
= 100
translations from Spanish to English by the
Answer the following questions:
5. Use Excel or SPSS to find a 90% confidence interval for the true mean BLEU score for translations by the Pharaoh program. Give your answer accurate to 4
decimal places. Interpret this confidence interval in a complete sentence.
(0.2832, 0.2920)
We are 90% confident that the true mean BLEU score for translations by the Pharaoh program lies between 0.2832 and 0.2920.
6. Use Excel or SPSS to find a 99% confidence interval for the true mean BLEU score for translations by the Pharaoh program. Give your answer accurate to 4
decimal places. Interpret this confidence interval in a complete sentence.
(0.2807, 0.2945)
(0.2807, 0.2945)
We are 99% confident that the true mean BLEU score for translations by the Pharaoh program lies between 0.2807 and 0.2945.
7. What do you notice about the confidence interval as the confidence level increased from 90% to 99%?
As the confidence level increases, the width of the confidence interval increases. We could also say, as the confidence level increases, the margin of error
increases. The center of the confidence interval (the sample mean) is unchanged.
4.2 Euro Weights
Summarize the relevant background information
A group of statisticians measured the weights of 2000 Belgian one Euro coins in eight batches. Each batch contains coins that were all minted together. You can
learn more about these data at: http://www.amstat.org/publications/jse/datasets/euroweight.txt
Describe the data collection procedures
The coins were "borrowed" from a bank in Belgium, one batch at a time. The weights (in grams) of the coins are given in the file EUROWEIGHT .
Answer the following questions:
8. Give the relevant summary statistics .
Excel
The following output was generated using the Excel file
QUANTITATIVEINFERENTIALP ROCEDURES.XLS:
SPSS
Using SPSS, we can find the summary statistics for these data:
Typically, we round our computations to one more decimal place of precision
than was given in the original data. This means we need one more decimal place
than is shown in the output above. We can increase the precision by selecting the
cell we want to modify and then clicking on the following button in the "Home"
menu ribbon in Excel:
This gives the following output:
When we do this, we get the following summary statistics:
x̄ = 7.5212
s = 0.0344
n = 2000
9. Make an appropriate graph to illustrate the data
Verify the requirements have been met
We need to check the following two requirements:
1. A simple random sample was drawn from the population
2. x̄ is normally distributed
The coins were taken in batches, but we can think of the batches as a random sample of the possible coins that could be chosen.
The data follow a bell-shaped distribution. There are a few potential outliers, but they are not too frequent or extreme. We can conclude that x̄ is normally
distributed.
Answer the following questions:
10. Find the confidence interval: Use Excel or SPSS to create a 99% confidence interval for the true mean.
(7.5193, 7.5232)
Answer the following questions:
11. Present your observations in an English sentence, relating the result to the context of the problem.
We are 99% confident that the true mean weight of all Belgian Euros is between 7.5193 grams and 7.5232 grams.
5 Summary
Remember...
In practice we rarely know the true standard deviation σ and will therefore be unable to calculate a z-score. Student's tdistribution gives us a new test statistic, t , that is calculated using the sample standard deviation (s ) instead.
x̄ − μ
t =
−
s/√n
The t -distribution is similar to a normal distribution in that it is bell-shaped and symmetrical, but the exact shape of the t distribution depends on the degrees of freedom (df ).
df = n − 1
You will use the appropriate software (Excel for 221 and SPSS for 222/223) to carry out hypothesis testing and create
confidence intervals involving t -distributions.
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