Download Example 8.11.4 (Using JMP to find a Confidence Interval for)

8.11 Using JMP
We turn now to the use of JMP for determining certain confidence intervals. The reader may
postpone the discussion below until covering the material for hypothesis testing in Chapter 9.
This postponement is helpful because JMP deals with confidence intervals and hypothesis testing
simultaneously.
Example 8.11.1 (Using JMP to Find a Confidence Interval for  ) Consider the following data
from a normal population with unknown mean  and unknown standard deviation  . Using
JMP, find a 95% confidence interval for the mean  and standard deviation  .
23
25
20
16
19
35
42
25
28
29
36
26
27
35
41
30
20
24
29
26
37
38
24
26
Solution: To find a 95% confidence interval for the mean  and standard deviation  using JMP,
proceed as follows:
1. From the main JMP taskbar, select File > New > Data Table, which results in an untitled
data table in a separate screen. To assign a title to this table, put the cursor in the box titled
“Untitled” in the left corner, right click twice, and enter the desired title.
2. In New Data Table enter the data in Column 1 and label it Data or Sample.
3. Select from the toolbar menu Analyze > Distribution. In the Select Column dialogue box,
select the column name where the data have been entered. With the column name highlighted,
select the Y, Columns box to the right under Cast Selected Columns into Roles. The column
name selected should populate the empty cell. Now, select OK. JMP now produces several
pieces of descriptive statistics. Select the red arrow icon in the box labeled with the data column
name and located just below the Distributions box at the very top of the output. A drop-down
menu appears as shown in Figure 8.11.1 that includes JMP’s various options for one population.
Figure 8.11.1 Pull-down menu showing JMP’s various options for one population.
4. From the pull-down menu in Figure 8.10.1, select Confidence Interval (or select Prediction
Interval if you want to predict a future observation) This allows you to choose the desired
confidence coefficient and one-sided or two-sided confidence intervals. Populate this cell with
the appropriate value and check the desired boxes.
5. After selecting OK, JMP supplements the existing output to include a 95% confidence interval
for the mean  and standard deviation  and a 95% prediction interval for a future observation
(if selected).
Table 8.11.1 JMP output of confidence intervals and
prediction interval for the data in Example 8.10.1.
6. To perform a z-test in JMP, follow the same steps as for the t-test. Then select Test mean
(see Chapter 9) from the pull-down menu in Figure 8.10.1. Enter the desired hypothesized value
of the mean in the box next to Specify Hypothesized Mean.
7. In the lower box, enter the true value or the estimated value of the standard deviation. If no
value of standard deviation is entered, the t-test is performed by default. We discuss this topic
further in Chapter 9, when we study hypothesis testing about the population mean.
Note that the JMP output in Table 8.10.1 also provides the confidence interval for the standard
deviation.
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Example 8.11.2 (Using JMP to find a Confidence Interval for 1  2 ) The following data give
the total cholesterol level among 10 women in each of two age groups.
Assume that the total cholesterol levels of these two segments of the population are normally
distributed with unknown means 1 and  2 and unknown variances  12 and  22 . Using JMP, find
a 95% confidence interval for the mean 2  1 .
Solution: To find a 95% confidence interval for the mean 2  1 using JMP, proceed as follows:
1. In New Data Table, enter the cholesterol level data in Column 1 and identifying labels in
Column 2. For example, designate the age groups 40-55 and 55-70 as 1 and 2, respectively.
Label Column 1 “cholesterol” (i.e., variable of interest) and label Column 2 “groups” or
“samples.”
2. Go to the Columns panel to the left of the datatable and right-click the blue triangle next to
column 2 (or groups). From the menu, check the option Nominal, since these data are nominal.
3. Select, from the main toolbar menu, Analyze > Fit Y by X. Then, in the new dialog box that
appears, select “cholesterol” as the Y variable and “groups” or “samples” as the X variable and
click OK.
4. In the resulting dialog box, select the red triangle icon next to Oneway Analysis of
cholesterol by Group box. A pull-down menu, as shown in Figure 8.10.2, appears that has all
the options JMP can perform.
Figure 8.11.2 Pull-down menu showing JMP’s options for two populations.
For example, if we assume that the variances of two populations are equal, then we select the
option Means/Anova/Pooled t. For the case of unequal variance, we choose t Test from the
pull-down menu. To change the confidence level, select Set  Level and enter the desired value
of  . Many of the other options in this pull-down menu are related to hypothesis testing, which
we shall discuss in Chapter 9. We have divided the JMP output into two parts, shown in Tables
8.10.2 and 8.10.3. The first part gives the confidence interval and hypothesis test for case of
equal variance, while the second part gives similar output for the case of unequal variance. We
shall revisit this example and discuss the hypothesis testing part in Chapter 9.
Table 8.11.2 JMP output for confidence interval and testing
of hypothesis for two population means with equal variances.
Table 8.11.3 JMP output for confidence interval and testing
of hypothesis for two population means with unequal variances.
Example 8.11.3 (Using JMP to find a Confidence Interval for  ) Refer to Example 8.7.1. A
random sample of 400 computer chips is taken from a large lot of chips and 50 of them are found
to be defective. Using JMP, find a 95% confidence interval for  , the proportion of defective
chips contained in the lot.
Solution: To find a 95% confidence interval using JMP for the proportion  of defective chips,
proceed as follows:
1. In a New Data Table, create two new columns. In Column 1, enter 0 for defective in the first
row and 1 for successful in the second row. Label Column 1 as results. Label Column 2 as
frequencies, and input the corresponding frequencies 50 and 350, respectively.
2. Go to the column panel to the left of the data table; right-click the blue triangle next to Results
and check the option Nominal, since these data are nominal.
3. Select from the bar menu Analyze > Distribution, then in the new dialog box that appears,
select Results as Y variable and Frequency as frequency variable, and click OK. A new dialog
box, Distribution of result, appears giving the point estimates of  and ( 1   ). Then, in this
dialog, select the red triangle icon next to Results. A pull-down menu containing various options
appears. From this pull-down menu, select the confidence interval option and then select the
desired confidence coefficient. The confidence intervals for  and (1   ) appear as shown in
Table 8.10.4.
Table 8.11.4 JMP output for point estimates
and confidence intervals for  and ( 1   ).
Example 8.11.4 (Using JMP to find a Confidence Interval for (1  2 ) ) Refer to Example 8.7.2.
Two companies, A and B, claim that their new types of light bulbs have a lifetime of more that
5,000 hours. In a random sample of 400 bulbs manufactured by company A, 60 bulbs burned out
before the claimed lifetime and in another random sample of 500 bulbs manufactured by
company B, 100 bulbs burned out before the claimed lifetime. Find a point estimate and a 95%
confidence interval for the true value of the difference (1  2 ) , where 1 and  2 are the
proportion of the bulbs manufactured by company A and company B, respectively, that burn out
before the claimed lifetime of 5,000 hours.
Solution: To find a 95% confidence interval using JMP for the difference of proportions (1  2 )
of defective bulbs, we proceed as follows:
1. To obtain a confidence interval for difference of two proportions, JMP uses the concept of a
2  2 contingency table (see Chapter 13). For instance, in this example the data in a 2  2
contingency table are entered in the format shown in Table 8.10.5.
2. Go to the column panel to the left of the data table and select a blue triangle next to the
Company and Results columns; right-click and then check the option Nominal, since the data in
theses columns are nominal.
3. Select from the bar menu Analyze > Fit Y by X. In the new dialog box, select Company,
Results, and Count as X, Y, and Frequency, respectively, and click OK. A new dialog Fit Y by
X appears. In this dialog, click the red triangle icon next to Contingency Analysis and then
select Two Sample Test for Proportions. The 95% confidence intervals the difference of
proportion of defectives and for the difference of proportion of non-defectives appear at the
bottom of this dialog under Two Sample Test for Proportions as shown in Table 8.10.5.
Table 8.11.5 JMP output for point estimates and confidence intervals
for (1  2 ) and [(1-1 )  (1-2 )  2  1 ].
Remark: Note that until Chapter 9, we postpone discussion of the technique for finding the
sample size, since for this we need to know the so-called power of the test.