Download Using your TI-Nspire Calculator: Estimating a Population Mean (σ

Using your TI-Nspire Calculator: Estimating a Population Mean (σ Unknown)
Dr. Laura Schultz
Statistics I
When the population standard deviation (σ) is not known (as is generally the case), a confidence
interval estimate for a population mean (μ) is constructed using a critical value from the relevant
Student’s t distribution. The TInterval calculator function will generate this confidence interval
using either raw sample data or summary statistics. Remember to confirm that the population is
normally distributed and/or n ≥ 30 before proceeding to generate any confidence intervals.
Generating a t interval from summary statistics:
1. Create a new document on your TI-Nspire and add a calculator.
2. Press the b key and select 6: Statistics followed by 6: Confidence Intervals. We’ll be using
option 2: t Interval.
3. To work with summary statistics, select Stats and press ·.
4. Consider the following example. An introductory statistics class at the U.S. Air Force Academy
counted how many chocolate chips were in each of 42 18-oz. bags of Chips Ahoy! cookies.
They found x = 1261.57 and s = 117.58 chocolate chips per bag. Without access to the actual
data, we’ll have to trust that it is safe to apply the Central Limit Theorem because n ≥ 30. Let’s
use the given summary statistics to construct a 95% confidence interval estimate of the mean (μ)
number of chocolate chips in all bags of Chips Ahoy! cookies. At the prompts, enter the sample
mean ( x ), sample standard deviation ( sx ), and the sample size (n). Enter 0.95 at the C-Level
prompt, and then highlight OK and press ·.
5. Your calculator will return the output screen shown above. Because we are working with
summary statistics, we would ordinarily round to the same number of decimal places as
Copyright © 2013 by Laura Schultz. All rights reserved.
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originally given for x . In this case, your calculator already rounded far enough. That’s okay;
you should worry about rounding only when your calculator reports more decimal places than
you started with for x .
6. How do we interpret this confidence interval? We are 95% confident that the interval from
1224.93 to 1298.21 actually does contain the true mean (μ) number of chocolate chips in all 18oz. bags of Chips Ahoy! cookies.
7. One way to construct a confidence interval estimate of μ is as x − E < µ < x + E , which would
be 1224.93 < μ <1298.21 chocolate chips per bag for this example.
8. We could also report the confidence interval as x ± E. We already know that x = 1261.57. We
can find the margin of error (E) the same way we did last week when we were working with
proportions. That is, we can use the formula
upper confidence limit - lower confidence limit
E=
, which gives us
2
1298.21− 1224.93
E=
= 36.64 for this example. (Note that the TI-Nspire reports the margin of
2
error as ME = 36.05). Hence, we could also construct this confidence interval as 1261.57 ±
36.64 chocolate chips per bag.
9. Go back and experiment with varying the confidence level (C-Level). What happens to the
width of the confidence interval when you use a 90% (0.90) confidence level? A 99% (0.99)
confidence level?
Generating a t interval from raw sample data:
1. Consider the following ages (in months) for a sample of 20 Rowan University Statistics I
students:
223
257
255
263
240
240
240
252
228
243
236
234
233
252
236
246
234
240
234
242
2. Press the ~ key and select 4: Insert followed by 6: Lists & Spreadsheets to add a list to your
TI-Nspire document. Name column A age and type in the above data. Be sure to check your
list for any typos.
Copyright © 2013 by Laura Schultz. All rights reserved.
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3. Recall that we can only apply the Central Limit Theorem if we know the population is normally
distributed and/or n ≥ 30. In this case, the sample size is not large enough, and we don’t know
whether the population of all Rowan Statistics I student ages is normally distributed. We can,
however, check whether our sample data are normally distributed. If so, then it is a pretty safe
bet that the population is also normally distributed. The easiest way to check is by plotting the
data distribution and deciding whether it “looks” normal. You could do this with a histogram,
box plot, or dot plot, but the most accurate method is to use a normal quantile plot. This
approach involves plotting the observed x values vs. the normal scores expected for a variable
that is normally distributed. If the resulting plot is a reasonably straight line, we can assume that
our population is normally distributed and proceed with generating a confidence interval.
4. Press the ~ key and select 4: Insert followed by 7: Data & Statistics to add a graphical
display to your TI-Nspire document. Click at the bottom of the screen and add the age variable
to the x-axis. The dots will dynamically rearrange themselves to form a dot plot. Note that this
dot plot is roughly symmetric and mound shaped, which is consistent with a normal distribution.
5. Press the b key and select 1: Plot Type followed by 4: Normal Probability Plot to produce a
normal quantile plot. For this example, the normal quantile plot is reasonably linear, so we are
safe to assume that the population consisting of the ages of all Rowan Statistics I students is
normally distributed. With the normality assumption satisfied, we can construct a 95%
confidence interval estimate of the mean age of all Rowan Statistics I students.
6. Go back to your calculator screen (or add a new one). Press the b key and select 6: Statistics
followed by 6: Confidence Intervals. We’ll be using option 2: t Interval again, but this time
select the Data option. Indicate that you will be using the age list and use a 95% confidence
level (C Level = 0.95).
Copyright © 2013 by Laura Schultz. All rights reserved.
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7. Your calculator will display the output screen shown above. Note that it reports both x (which
is the best point estimate of μ) and the sample standard deviation ( sx ) in addition to the
confidence interval limits. Because we used the original sample data (as opposed to summary
statistics), round the confidence interval limits to one more decimal place than we had for the
raw data. For this example, we could construct a 95% confidence interval estimate for the mean
(μ) age of all Rowan Statistics I students as either 236.6 < μ < 246.2 months or 241.4 ± 4.8
months.
8. How do we interpret this confidence interval? We are 95% confident that the true mean age of
all Rowan Statistics I students is somewhere between 236.6 and 246.2 months.
Copyright © 2013 by Laura Schultz. All rights reserved.
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