Inferential Statistics Utilities for TI-83/84

Examples

Example 1: A machine packs cereal into boxes, and you require a standard deviation of no more than five grams. You randomly select and weigh 45 boxes and find a sample standard deviation of 6.2 grams. Is the machine operating within specification?

You have tested the sample and find that it is normally distributed with no outliers, so you are confident that the population is also normally distributed.

Solution: n = 45, s = 6.2, σ_o = 5. Your hypotheses are

H₀: σ ≤ 5, the machine is within spec (some books would say H₀: σ=5)

H₁: σ > 5, the machine is not working right

No α was specified, but for an industrial process with no possibility of human injury α = 0.05 seems appropriate.

Run the MATH200I program and select 2:Infer about σ. Enter s:6.2 and n:45, and select 5:Test σ>const. Enter 5 for σ in H₀.

The results are shown at far right. The test statistic is χ²_o = 67.65 with 44 degrees of freedom, and the p-value is 0.0125.

Since p<α, you reject H₀ and accept H₁. At the 0.05 level of significance, the population standard deviation σ is greater than 5, and the machine is not operating within specificaton.

Example 2: You have a random sample of size 20, with a standard deviation of 125. You have good reason to believe that the underlying population is normal, and you’ve checked the sample and found no outliers. Is the population standard deviation different from 100, at the 0.05 significance level?

Solution: n = 20, s = 125, σ_o = 100, α = 0.05. Your hypotheses are

H₀: σ = 100

H₁: σ ≠ 100

This time in the INFER ABOUT σ menu you select 4:Test σ≠const.

Results are shown at right. χ²_o = 29.69 with 19 degrees of freedom, and the p-value is 0.1118.

p>α; fail to reject H₀. At the 0.05 significance level, you can’t say whether the population standard deviation σ is different from 100 or not.

Example 3: Of several thousand students who took the same exam, 40 papers were selected randomly and statistics were computed. The standard deviation of the sample was 17 points. Estimate the standard deviation of the population, with 95% confidence. (Recall that test scores are normally distributed.)

Solution: Check the data and make sure there are no outliers. Run MATH200I and select [2] in the first menu. Enter s and n, and in the second menu select 1:σ interval with a C-level of 95 or .95. The results screen is shown at right.

Conclusion: You’re 95% confident that the standard deviation of test scores for all students is between 13.9 and 21.8.

Remark: The center of the confidence interval is about 17.9, which is different from the point estimate s=17. This is a feature of confidence intervals for σ or σ²: they are asymmetric because the χ² distribution used to compute them is asymmetric.

Example 4: Heights of U.S. males aged 18–25 are normally distributed. You take a random sample of 100 from that population and find a mean of 65.3 in and a variance of 7.3 in². (Remember that the units of variance are the square of the units of the original measurement.)

Estimate the mean and variance of the height of U.S. males aged 18–25, with 95% confidence.

Solution for mean: Computing a confidence interval for the mean is a straightforward TInterval. Just remember that for Sx the calculator wants the sample standard deviation, but you have the sample variance, which is s². Therefore you take the square root of sample variance to get sample standard deviation, as shown in the input screen at near right.

The output screen at far right shows the confidence interval. You’re 95% confident that the mean height of U.S. males aged 18–25 is between 64.8 and 65.8 in.

Solution for variance: Run the MATH200I program and select 2:Infer about σ. Enter s:√7.3 and n:100. Select 2:σ² interval and enter C-Level:.95 (or 95). The program computes the confidence interval for population variance as 5.6 ≤ σ² ≤ 9.9. Notice that the output screen shows the point estimate for variance, s², and that as expected the confidence interval is not symmetric.

You’re 95% confident that the variance in heights of U.S. males aged 18–25 is between 5.6 and 9.9 in².

Complete answer: You’re 95% confident that the heights of U.S. males aged 18–25 have mean 64.8 to 65.9 in and variance 5.6 to 9.9 in².

3. Goodness-of-Fit Test

The goodness-of-fit or GoF test determines how well a multinomial model (more than two categories) matches the sample data. You have to sum up the χ² contributions of all the cells. Those are found by using the formula

(Observed − Expected)² / Expected

where the Expected numbers themselves are found by

Model × ∑Observed / ∑Model

The TI-83 does none of those computations; the TI-84 computes χ² contributions but only after you compute the Expected numbers. The MATH200I program does all the computations.

Before running the MATH200I program, put your model in L1 and your Observed numbers in L2. Then run MATH200I and select 3:GoF test. The program will ask you to confirm that you’ve filled the two lists, and then it will perform all computations for the χ² goodness of fit and show the results on a graph. The Expected numbers are left in L3, and you should verify that they meet the requirements for a χ² test: none of them <1, and no more than 20% of them <5.

Example 1 from Sullivan, Michael, Fundamentals of Statistics (Pearson Prentice Hall, 2008), Example 3, pages 555–556.

“An urban economist wonders if the distribution of residents in the United States is different today than it was in 2000. In 2000,” the proportions were as shown in the table. “The economist randomly selects 1500 households and obtains the frequency distribution shown. ... Conduct the appropriate test to determine if the distribution of residents in the United States is different today from the distribution in 2000, using the α = 0.05 level of significance.”

Solution: H₀ is that the model is still good, and H₁ is that the distribution of U.S. population has changed. The model is the percentages shown, and you are determining whether the current sample is enough different from the model for you to conclude that the model is no longer correct.

Put the percentages in L1; there’s no need to convert them to decimals but if you do then you must convert all of them. The current sample is the Observed numbers, and they go in L2. Run the MATH200I program, select 3:GoF test, and press [9] to confirm that you have data in the correct lists.

Caution: The Observed numbers must always be actual counts; never convert them to percentages.

Caution: Some problems will show totals, but you never enter the totals in your calculator.

The results screen is shown at right. The χ² test statistic is 8.25, with three degrees of freedom (four categories minus one). The p-value is 0.0410.

Conclusion: Since p<α, you reject H₀ and accept H₁. At the 0.05 level of significance, you conclude that the regional distribution of U.S. residents is different today from what it was in 2000.

Note: Your book rounds its calculations at various stages, so its χ² test statistic may be slightly less accurate than yours. Your book also uses tables to look up p-values, where your calculator uses a more accurate method of computation. So don’t worry if your numbers don’t match the book’s exactly.

The screen also reminds you to check L3, the Expected numbers, to make sure that the requirements for a χ² test are met. In fact, there’s useful information in L3 through L5, as shown at right.

L3 contains the expected numbers. Always check them after the computation to make sure that none of them are below 1 and no more than 20% of them are below 5. Here, all are well above 5.

L4 is the absolute differences between Observed and Expected. For instance, in the first category the Observed number was 11 below what would be expected if H₀ is exactly true. Of course, it’s highly unusual for Observed numbers to match Expected numbers exactly. The question is how far off the numbers are, taking into account the weights of Expected numbers in the model. That’s what L5 shows, namely each category’s contribution to χ². You can see that the most important deviation is in the second category (4.78), and the least important is in the first category (0.42). The total of L5 is the χ² test statistic.

Example 2—equal frequencies from Sullivan, Michael, Fundamentals of Statistics (Pearson Prentice Hall, 2008), Example 4, pages 556–558.

“An obstetrician wants to know whether or not the proportion of children born each day of the week is the same. She randomly selects 500 birth records and obtains the data shown.” Clearly the frequencies in the sample vary from day to day, but do they vary enough that you can say babies in general are born with different frequencies on different days, at the 0.05 level of signifcance?

Solution: H₀ is that babies are born with equal frequency on all days of the week, and H₁ is that they are not. You’re not given a numerical model because “equal frequencies” means that all model numbers are the same. Since there are seven categories, your model is seven 1’s in L2. The Observed numbers go in L2, as usual.

The results are shown at right. Here df = 6 because there are seven categories. The χ² test statistic is 6.18, and the p-value is 0.4029. That is greater than α, and so you fail to reject H₀.

Writing non-conclusions is problematic when p>α in a χ² test. Strictly speaking, the conclusion should be in neutral language as usual: you can’t determine from the data whether babies are born with equal frequency on all the days of the week or not. But traditionally, the conclusion is often stated in some words equivalent to “the data do not rule out the model”.

This is the scientific method. If the experiment is repeated multiple times and H₀ is never rejected, we begin to have more and more confidence that H₀ is actually true. We can’t accept H₀ from a single experiment, but the more times it’s not rejected, the more we believe that it may never be rejected.

4. Inferences about Linear Correlation

With linear correlation, you compute a sample correlation coefficient r. But what can you say about the correlation in the population, ρ? The MATH200I program computes a confidence interval about ρ or performs a hypothesis test to tell whether there is correlation in the population.

See also:  Inferences about Linear Correlation gives the statistical concepts with examples of calculation “by hand” and in an Excel workbook.

To perform inferences about linear regression, first load your x’s and y’s in any two statistics lists. Then run the MATH200I program and select 5:Regression inf.

Example: The following sample of commuting distances and times for fifteen randomly selected co-workers is adapted from Johnson & Kuby Elementary Statistics (Thomson, 2004), page 623.

The TI’s LinReg(ax+b) command can tell you that the correlation of the sample is 0.88. But what can you infer about the correlation of the population?

Confidence Interval about ρ

Enter your x’s and y’s in two statistics lists, such as L1 and L2. Run the MATH200I program and select 4:Correlatn inf. When prompted, enter your x list and y list, and your desired confidence level, such as .95 or 95 for 95%.

The output screen is shown at right. For this sample of n = 15 points, the sample correlation coefficient is r = 0.88. For the correlation of the population (distances and times for all commuters at this company), you’re 95% confident that 0.67 ≤ ρ ≤ 0.96.

Hypothesis Test about ρ

You can also do a hypothesis test to see whether there is any correlation in the population. The null hypothesis H₀ is that there is no correlation in the population, ρ = 0; the alternative H₁ is that there is correlation in the population, ρ ≠ 0.

Select your α; 0.05 is a common choice. Run the MATH200I program and select 4:Correlatn inf. Enter your x and y lists and enter 0 for C-Level; this tells the program that you want a hypothesis test.

The output screen is shown at right. Sample size n = 15, and sample correlation is r = 0.88. The t statistic for this hypothesis test is 6.64, and with 13 (n−2) degrees of freedom that yields a p-value of <0.0001.

p<α; reject H₀ and accept H₁. At the 0.05 level of significance, ρ ≠ 0: there is some correlation in the population. Furthermore, the population correlation is positive. (See p < α in Two-Tailed Test: What Does It Tell You? for interpreting the result of a two-tailed test in a one-tailed manner like this.)

Remark: When p is greater than α, you fail to reject H₀. In that case, you would conclude that it is impossible to say, at the 0.05 level of significance, whether there is correlation in the population or not.

5. Inferences about Linear Regression

A linear regression fits an equation of the form ŷ = b₁x + b₀ to the sample data, but the slope b₁ and the y intercept b₀ are just sample statistics. If you took a different sample you would likely get a different regression line.

The MATH200I program finds confidence intervals for the slope β₁ and intercept β₀ of the line that best fits the entire population of points, not just a particular sample. It can also find a confidence interval about the mean ŷ for a particular x and a prediction interval about all ŷ’s for a particular x.

The program doesn’t do any hypothesis tests on the regression line. The standard test is to test whether the regression line has a nonzero slope, β₁ ≠ 0. But that test is identical to the test for a nonzero correlation coefficient, ρ ≠ 0, which the MATH200I program performs as part of the 4:Correlatn inf menu selection.

See also:  Inferences about Linear Regression explains the principles and calculations behind inferences about linear regression; there’s even an Excel workbook.

To perform inferences about linear regression, first load your x’s and y’s in any two statistics lists. Then run the MATH200I program and select 5:Regression inf.

Example: Let’s use the same data on commuting distances and times that you used for Inferences about Linear Correlation.

The TI-83/84 command LinReg(ax+b)will show the best fitting regression line for this particular sample, but what can we say about the regression for all commuters at that company?

Regression Coefficients for the Population

Solution: Enter the x’s and y’s in any two statistics lists, such as L1 and L2. Run the MATH200I program and select 5:Regression inf. Specify the two lists and your desired confdence level, such as .95 or 95 for 95%.

Results: The slope of the sample regression line is 1.89, meaning that on average each extra mile of commute takes 1.89 minutes (a speed of about 31 mph). But the 95% confidence interval for the slope is 1.28 to 2.51: you’re 95% confident that the slope of commuting time per distance, for all commuters at this company, is between 1.28 and 2.51 minutes per mile (46 and 23 mph).

The second section of the screen shows that the y intercept of the sample is 3.6: this represents the “fixed cost” of the commute, as opposed to the “variable cost” per mile represented by the slope. But the 95% confidence interval is −4.5 to +11.8 minutes.

Interpretation: the line that best fits the sample data is

ŷ = 1.89x + 3.6

and the regression line for the whole population is

ŷ = β₁x + β₀

where you’re 95% confident that

1.28 ≤ β₁ ≤ 2.51   and   −4.5 ≤ β₀ ≤ +11.8

Let’s think a bit more about that intercept, with a 95% confidence interval of −4.5 to +11.8 minutes. This is a good illustration that it’s a mistake to use a regression line too far outside your actual data. Here, the x’s run from 3 to 20. The y intercept corresponds to x = 0, and a commute of zero miles is not a commute at all. (Yes, there are people who work from home, but they don’t get in their cars and drive to work.) While the y intercept can be discussed as a mathematical concept, it really has no relevance to this particular problem.

Inferences about a Particular x Value

The first output screen was about the line as a whole; now the program turns to predictions for a specific x value. First it asks for the x value you’re interested in. This time, let’s make predictions about a commute of 10 miles.

Caution: You should only use x values that are within the domain of x values in your data, or close to it. No matter how good the straight-line relationship of your data, you don’t really know whether that relationship continues for lower or higher x values.

The program arbitrarily limits you to the domain plus or minus 15% of the domain width, but even that may be too much in some problems. In this problem, commuting distances range from 3 mi to 20 mi, a width of 17 mi. The program will let you make predictions about any x value from 3−.15*12 = 0.45 mi to 15+.15*12 = 22.55 mi, but you have to decide how far you’re justified in extrapolating.

The input and output screens are shown at right. ŷ (y-hat) is simply the y value on the regression line for the given x value, found by ŷ = (slope)×10+(intercept) = 22.6. That is a prediction for μ_y|x=10, the average time for many 10-mile commutes. The screen shows a 95% confidence interval for that mean: you’re 95% confident that the average commute time for all 10-mile commutes (not just in the sample) is between 19.3 and 25.9 minutes.

But that is an estimate of the mean. Can we say anything about individual commutes? Yes, that is the prediction interval at the bottom of the screen. It says that 95% of all 10-mile commutes take between 10.4 and 34.7 minutes.

6. Critical t or Inverse t

The TI-83 doesn’t have an invT function as the TI-84 does, but if you need to find critical t or inverse t on either calculator you can use this part of the MATH200I program.

Caution: our notation of t(df,rtail) matches most books in specifying the area of the right-hand tail for critical t. But the TI calculator menus specify the area of the left-hand tail. Make sure you know whether you expect a positive or negative t value.

Some textbooks interchange the arguments: t(rtail,df). Since degrees of freedom must always be a whole number and the tail area must always be less than 1, you’ll always know which argument is which.

Example: find t(27,0.025), the t statistic with 27 degrees of freedom (sample size 28) for a one-tailed significance test with α = 0.025, a two-tailed test with α = 0.05, or a confidence interval with 1−α = 95%.

Solution: run the MATH200I program and select 6:Critical t. When prompted, enter 27 for degrees of freedom and 0.025 for the area of the right-hand tail, as shown in the first screen. After a short pause, the calculator gives you the answer: t(27,0.025) = 2.05.

Interpretation: with a sample of 28 items (df=27), a t score of 2.05 cuts the t distribution with 97.5% of the area to the left and 2.5% to the right.

7. Critical χ² or Inverse χ²

χ²(df,rtail) is the critical value for the χ² distribution with df degrees of freedom and probability rtail. (In the context of a hypothesis test, rtail is α, the significance level of the test.)

In the illustration, rtail is the area of the right-hand tail, and the asterisk * marks the critical value χ²(df,rtail). The critical value or inverse χ² is the χ² value such that a higher value of χ² has only an rtail probability of occurring by chance.

You can compute critical χ² only for the right-hand tail, because the χ² distribution has no left-hand tail.

Caution:  Some textbooks write the function the other way, χ²(rtail,df). Since df is a whole number and rtail is a decimal between 0 and 1, you will be able to adapt.

Example: What is the critical χ² for a 0.05 significance test with 13 degrees of freedom?

Run the MATH200I program and select 7:Critical χ². Enter the number of degrees of freedom and the area of the right-hand tail. Be patient: the computation is slow. But the program gives you the critical χ² value of 22.36, as shown in the second screen.

Interpretation: For a χ² distribution with 13 degrees of freedom, the value χ² = 22.36 divides the distribution such that the area of the right-hand tail is 0.05.

Reference: Variable Usage

Most parts of the MATH200I program leave useful results in variables, which you can use for further calculation. Use the [ALPHA] key. For instance, if you want a value from variable V, press [ALPHA 6 makes V]. Also, if you’re using any variables or statistics lists yourself, you don’t want to be surprised when the program changes their values. Below you’ll find complete information.

1:Sample size

• C = confidence level (adjusted to >0, <1 if necessary)
• E = margin of error
• N = required sample size, a whole number
• P = prior estimate p̂ or p̂₁
• Q = prior estimate p̂₂, or 0
• S = standard deviation of population or sample
• Z = critical t(df,α/2) or z(α/2), unrounded
• used by program, not useful afterward: M, Str0 (deleted), Str1 (deleted), X

2:Infer about σ

• C = confidence level (adjusted to >0, <1 if necessary)
• D = degrees of freedom = N−1
• H = high end of confidence interval for σ or σ²
• L = low end of confidence interval for σ or σ²
• N = sample size
• P = p-value, unrounded
• S = standard deviation of sample
• Z = σ₀ from H₀
• used by program, not useful afterward: Str0 (deleted), X

3:GoF test

• D = degrees of freedom (list length minus 1)
• L3 = Expected numbers
• L4 = deviations, Observed−Expected
• L5 = χ² contributions, (Observed−Expected)²/Expected
• P = p-value, unrounded
• X = χ² test statistic

4:Correlatn inf

• C = confidence level, or 0 for hypothesis test
• H = high end of confidence interval for ρ
• L = low end of confidence interval for ρ
• N = sample size
• P = p-value for hypothesis test, unrounded
• R = sample correlation coefficient
• T = test statistic for hypothesis test, unrounded
• Z = Fisher Z for confidence interval
• LX = x list (deleted)
• LY = y list (deleted)

5:Regression inf

• A = slope of sample regression line
• B = y intercept of sample regression line
• C = confidence level (adjusted to >0, <1 if necessary)
• M = x̄
• N = sample size
• X = x value entered by user
• Y = ŷ value for X
• LX = x list (deleted)
• LY = y list (deleted)
• used by program, not useful afterward: E, F, S, Z, GDB0 (deleted)

6:Critical t

• A = area of right-hand tail
• D = degrees of freedom
• T = critical t value

7:Critical χ²

• A = area of right-hand tail
• D = degrees of freedom
• X = critical χ² value

What’s New

14 Nov 2009: Add note about future replacement of this program

18 Oct 2009: several small edits for clarity and to correct typos

24 Aug 2009: Protect the program to prevent accidental editing.

8 Mar 2009: In 4:Correlatn inf in the program, check for r = ±1, which prevents doing inferences. Only the program changed; there were no changes needed in this document.

7 Dec 2008: Add a program version that works with the original TI-83.

29 Nov 2008: several changes in the MATH200I program:

• change Lbl statements so that the program will run on an original TI-83
• optimize Text statements where values follow on the same line
• put numeric cases first in the menu for required sample size
• round p-values to four decimals and show “<.0001” instead of “0”
• round critical t and z in sample-size calculations to two decimals

24 Nov 2008:

• pull five separate programs into a unified MATH200I application, and six Web pages into this one, eliminating a fair amount of duplication; document variable usage clearly
• add Determining Necessary Sample Size and Critical t or Inverse t
• in the MATH200I program, add titles to most program screens, and change prompts to lower case or mixed case for readability
• in Inferences about σ, point out that the confidence intervals aren’t symmetric and explain why
• in Goodness-of-Fit Test, use new examples from the textbook; in the 3:GoF test part of the MATH200I program, check that L1 and L2 have equal length and improve the captions of numbers on the results graph
• in the 4:Correlatn inf part of the MATH200I program, use input x and y lists instead of asking the user for r and n; save results in L and H
• in the 5:Regression inf part of the MATH200I program, improve formatting and reduce the four output screens to two, and check the entered x value for reasonableness
• In the 7:Critical χ² part of the MATH200I program, add error checking

Here’s part of the earlier history of the pages now in this page and the programs now part of the MATH200I program:

1. Determining Necessary Sample Size and its part of the program were created on 23 Nov 2008 and were the impetus for drawing all the utilities for inferential stats together to escape keeping track of a bunch of small programs.
2. Inferences about σ and its part of the program were created on 9 Aug 2008 as Inferences about One-Pop. Standard Deviation on TI-83/84 and program SDINFER.
3. Goodness-of-Fit Test was created on 23 Jun 2002 as Multinomial Experiments, and its part of the program was created on 2 May 2004 as MULTINOM.
4. Inferences about Linear Correlation and its part of the program were created on 4 Apr 2008 as Inferences about Linear Correlation on TI-83/84/89 and program CORINFER.
5. Inferences about Linear Regression and its part of the program were created on 4 Apr 2008 as Inferences about Linear Regression on TI-83/84 and program REGINFER.
6. Critical t or Inverse t was created on 30 Mar 2003 as a description of the TInterval “back door” called Computing Inverse t. This part of the MATH200I program was created on 23 Nov 2008 and there never was a separate program.
7. Critical χ² or Inverse χ² and its part of the program were created on 19 Aug 2007 as Computing Critical χ² on TI-83/84/89 and INVCHI2.