Chapter 8
Technology
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Example T.1: Using a TI-84 Plus Calculator to Find the
Value of t Given the Area to the Left
Find the value of t for a t-distribution with 17 degrees
of freedom such that the area under the curve to the
left of t is 0.10.
Solution
Press
and then
to go to the DISTR menu.
Choose option 4:invT(. The function syntax for this
option is invT(area to the left of t, df). Since the area
to the left is the given area in this example, we would
enter invT(0.10,17). Press
.
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Example T.1: Using a TI-84 Plus Calculator to Find the
Value of t Given the Area to the Left (cont.)
The answer given by the calculator is t  −1.333.
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Example T.2: Using a TI-83/84 Plus Calculator to Find a
Confidence Interval for a Population Mean (s Known)
Use a TI-83/84 Plus calculator to find an 85%
confidence interval for the population mean amount of
money that teachers spend on lunch each week. A
survey of 42 randomly selected teachers found that
they spend a mean of $18.00 per week on lunch.
Assume that the population standard deviation is
known to be $2.00 per week.
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Example T.2: Using a TI-83/84 Plus Calculator to Find a
Confidence Interval for a Population Mean (s Known) (cont.)
Solution
To begin, press
; then scroll over and choose
TESTS. From the TESTS menu, choose option
7:ZInterval. Choose Stats, and then press
. Enter 2 for s, 18 for Ë, a sample size of 42 for
n, and 0.85 for the level of confidence, C-Level.
Highlight Calculate, and then press
.
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Example T.2: Using a TI-83/84 Plus Calculator to Find a
Confidence Interval for a Population Mean (s Known) (cont.)
The screen will display the results shown in the
screenshot.
Thus we are 85% confident that the mean amount of
money that teachers spend on lunch each week is
between $17.56 and $18.44.
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Example T.3: Using a TI-83/84 Plus Calculator to Find a
Confidence Interval for a Population Mean (s Unknown)
Use a TI-83/84 Plus calculator to find a 95% confidence
interval for the population mean amount of money that
teachers spend on lunch each week. A survey of 12
randomly selected teachers found that they spend a
mean of $20.00 per week on lunch with a sample
standard deviation of $3.00 per week.
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Example T.3: Using a TI-83/84 Plus Calculator to Find a
Confidence Interval for a Population Mean (s Unknown) (cont.)
Solution
To begin, press
; then scroll over and choose
TESTS. From the TESTS menu, choose option
8:TInterval. Choose Stats, and then press
. Enter 20 for Ë, 3 for Sx, a sample size of 12
for n, and 0.95 for the level of confidence, C-Level.
Highlight Calculate, and then press
.
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Example T.3: Using a TI-83/84 Plus Calculator to Find a
Confidence Interval for a Population Mean (s Unknown) (cont.)
The screen will display the results shown in the
screenshot.
Thus, we are 95% confident that the mean amount of
money that teachers spend on lunch each week is
between $18.09 and $21.91.
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Example T.4: Using a TI-83/84 Plus Calculator to Find a
Confidence Interval for a Population Proportion
A survey of 345 randomly selected students at one
university found that 301 students think that there is
not enough parking on campus. Find the 90%
confidence interval for the proportion of all students at
this university who think that there is not enough
parking on campus.
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Example T.4: Using a TI-83/84 Plus Calculator to Find a
Confidence Interval for a Population Proportion (cont.)
Solution
Press
; then scroll over and choose TESTS. From
the TESTS menu, choose option A:1-PropZInt. Of
those surveyed, 301 students think that there is not
enough parking, so enter 301 for x. Our sample size is
345, so enter 345 for n. Our level of confidence is 90%.
This must be entered as a decimal, so enter 0.90 for
C-level. Choose Calculate and press
.
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Example T.4: Using a TI-83/84 Plus Calculator to Find a
Confidence Interval for a Population Proportion (cont.)
The screen will display the results shown in the
screenshot.
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Example T.4: Using a TI-83/84 Plus Calculator to Find a
Confidence Interval for a Population Proportion (cont.)
Notice that the confidence interval is displayed as well
as the value of the sample proportion, p̂. Thus, the 90%
confidence interval for the proportion of all students at
this university who think that there is not enough
parking on campus is (0.843, 0.902).
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Example T.5: Using Microsoft Excel to Find the
Value of tα
Find the value of t0.05 for the t-distribution with 15
degrees of freedom.
Solution
We know the number of degrees of freedom is 15. The
area, 0.05, is the area to the right of t, which is the area
in one tail of the distribution. Using Microsoft Excel
2010, enter =T.INV(0.05, 15). The value displayed will
be approximately -1.753, since the formula assumes
that we are entering the area in the left tail. By
changing the resulting value of t to a positive number,
we obtain t0.05 ≈ 1.753.
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Example T.6: Using Microsoft Excel to Find the
Value of t Given Area between -t and t
Find the critical t-value for a t-distribution with 29
degrees of freedom such that the area between −t and
t is 99%.
Solution
We know the number of degrees of freedom is 29.
Since 99% of the area of the curve is in the middle, that
leaves 1%, or 0.01 of the area in the two tails. Thus, we
need to use the formula for area in two tails, with an
area, or probability, of 0.01.
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Example T.6: Using Microsoft Excel to Find the
Value of t Given Area between -t and t (cont.)
Unlike other formulas we have used that required us to
divide that area by 2, this formula has that calculation
built in. Using Microsoft Excel 2010, enter
=T.INV.2T(0.01, 29). The value displayed will be
approximately 2.756. This is the positive t-value. Thus,
99% of the area lies between -t ≈ -2.756 and t ≈ 2.756.
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Example T.7: Using Microsoft Excel to Find the Margin of Error
for a Confidence Interval for a Population Mean (s Known)
Calculate the margin of error for a 95% confidence
interval for a population mean given that s = 8.5 and
n = 47.
Solution
Since the value for the population standard deviation is
given, we want to use a normal distribution to calculate
the margin of error. Remember that a = 1 − c,
so a = 1 − 0.95 = 0.05. In Excel 2010, enter
=CONFIDENCE.NORM(0.05, 8.5, 47). Thus, the margin
of error is approximately 2.430066.
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Example T.8: Using Microsoft Excel to Find the Margin of Error for a
Confidence Interval for a Population Mean (s Unknown) (cont.)
Calculate the margin of error for a 95% confidence
interval for a population mean given that s = 8.5 and
n = 47. Use a t-distribution.
Solution
Since the value for the population standard deviation is
unknown, it is appropriate to use a t-distribution. We
are told to use this distribution as well. Remember that
a = 1 − c, so a = 1 − 0.95 = 0.05. In Excel 2010, enter
=CONFIDENCE.T(0.05, 8.5, 47). Thus, the margin of
error is approximately 2.495693.
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Example T.9: Using Microsoft Excel to Construct a
Confidence Interval for a Population Proportion
During the 2010–2011 NBA season, Kobe Bryant
attempted 583 free throws and made 483 of these.
Construct a 99% confidence interval for his true free
throw percentage.
Source: FOX Sports on MSN. “NBA Sortable Stats.” 2012.
http://msn.foxsports.com/nba/sortableStats?league=NBA&dir=descending&s
tat=&low=1&high=50&table=freethrows&position=all&showPlayers=min&ye
ar=2010&seasonState=regular&Go=Go (22 Jan. 2012).
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Example T.9: Using Microsoft Excel to Construct a
Confidence Interval for a Population Proportion (cont.)
Solution
The first step is calculating a point estimate for the
interval. Enter the number of successes, 483, into the
cell B1 and the total attempts, 583, into B2. We divide
the number of successes by the total attempts. Enter
=B1/B2 to calculate p.̂ The margin of error is calculated
by entering =NORM.S.INV(0.995)*SQRT(B3*(1B3)/B2). Finally, subtracting the margin of error from p̂
yields the lower endpoint of the interval and adding the
margin of error to p̂ yields the upper endpoint.
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Example T.9: Using Microsoft Excel to Construct a
Confidence Interval for a Population Proportion (cont.)
Thus, based on his performance during the 2010–2011
NBA season, we are 99% confident that Kobe Bryant’s
true free throw percentage is between 78.8% and
86.9%.
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Example T.10: Using Minitab to Find a Confidence
Interval for a Population Mean (s Unknown)
Given the following sample data from a study on the
average amount of water used per day by members of
a household while brushing their teeth, calculate the
99% confidence interval for the population mean using
Minitab. Assume that the sample used in the study was
a simple random sample.
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Example T.10: Using Minitab to Find a Confidence
Interval for a Population Mean (s Unknown) (cont.)
Household Water Used for Brushing Teeth (in Gallons per Day)
0.485
0.587
0.412
0.942
0.868
0.925
0.805
0.515
0.428
0.516
0.367
0.943
0.898
0.95
0.810
0.463
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0.39
0.465
0.336
0.940
0.889
0.959
0.839
0.420
0.308
0.370
0.269
0.941
0.910
0.948
0.860
0.326
0.231
0.282
0.198
0.946
0.927
0.956
0.861
0.243
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Example T.10: Using Minitab to Find a Confidence
Interval for a Population Mean (s Unknown) (cont.)
Solution
Since we are not told any population parameters for
the study, we cannot assume that s is known or that
the population distribution is approximately normal.
However, since the sample is large enough, we can use
the t-distribution to construct a confidence interval for
the population mean. Begin by typing the sample data
into the first column, C1.
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Example T.10: Using Minitab to Find a Confidence
Interval for a Population Mean (s Unknown) (cont.)
Next, to produce the dialog window for finding a
confidence interval for a population mean using a
t-distribution, go to Stat ► Basic Statistics ► 1-Sample
t. Next, select Samples in columns and enter C1 in the
box. Then select Options to enter 99 for the confidence
level. Click OK on the options window and OK on the
main dialog window and your results will be displayed
in the Session window.
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Example T.10: Using Minitab to Find a Confidence
Interval for a Population Mean (s Unknown) (cont.)
HAWKES LEARNING SYSTEMS
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Example T.10: Using Minitab to Find a Confidence
Interval for a Population Mean (s Unknown) (cont.)
HAWKES LEARNING SYSTEMS
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Example T.10: Using Minitab to Find a Confidence
Interval for a Population Mean (s Unknown) (cont.)
The results in the Session window should appear as
follows.
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Example T.10: Using Minitab to Find a Confidence
Interval for a Population Mean (s Unknown) (cont.)
The third column of the results displays the point
estimate, x,̄ and the last column contains the
confidence interval. The 99% confidence interval is
(0.5240, 0.7624). Therefore, we are 99% confident that
the mean amount of water used per household for
brushing teeth is between 0.5240 and 0.7624 gallons
per day.
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Example T.11: Using Minitab to Find a
Confidence Interval for a Population Proportion
A certain dosage of insecticide is applied to a sample of
200 cockroaches in a laboratory. The scientists have
predicted that this dose should kill more than 50% of
the cockroaches. When the results are tallied, 83
cockroaches are dead. Construct a 95% confidence
interval for the percentage of cockroaches killed by the
insecticide and use the interval to determine whether
the evidence supports the scientists’ claim.
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Example T.11: Using Minitab to Find a Confidence
Interval for a Population Proportion (cont.)
Solution
To produce the dialog window for a confidence interval
for a population proportion, go to Stat ► Basic
Statistics ► 1 Proportion. Next, select Summarized
data and enter 83 for the number of events and 200 for
the number of trials. Then select Options to enter 95
for the confidence level and select the option, Use test
and interval based on normal distribution. Click OK on
the options window and OK on the main dialog window
and your results will be displayed in the Session
window.
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Example T.11: Using Minitab to Find a Confidence
Interval for a Population Proportion (cont.)
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Example T.11: Using Minitab to Find a Confidence
Interval for a Population Proportion (cont.)
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Example T.11: Using Minitab to Find a Confidence
Interval for a Population Proportion (cont.)
The results in the Session window should appear as
follows.
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Example T.11: Using Minitab to Find a Confidence
Interval for a Population Proportion (cont.)
The fourth column displays the point estimate, p,̂ and
the fifth contains the confidence interval. The 95%
confidence interval, (0.346713, 0.483287), does not
contain any values greater than 0.50. Therefore, the
scientists’ claim that the tested dosage of insecticide
should kill more than 50% of the cockroaches is not
supported at a confidence level of 95%.
HAWKES LEARNING SYSTEMS
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