Using your TI-Nspire Calculator: Estimating a Population...

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

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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 18-oz. 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

E = upper confidence limit - lower confidence limit2

, which gives us

E = 1298.21−1224.932

= 36.64 for this example. (Note that the TI-Nspire reports the margin of

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.


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).


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.


Using your TI-Nspire Calculator: Estimating a Population...

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