Appendix D — Manual Graphing Calculators

D.1 Graphing Calculator Texas Instruments TI-83/TI-83 Plus/TI-84 Plus

D.1.1 Measures of central tendency and variability

Choose: STAT > EDIT

  • Enter the outcomes in list L1, or
  • Enter the different outcomes in list L1 and the corresponding frequencies in list L2
  • Leave L1 (or L1, and L2) via 2nd MODE (= QUIT)

Choose: STAT > CALC > 1-VAR STATS > List: L1, or STAT > CALC > 1-VAR STATS > List: L1, and FreqList: L2

You will get, among other values:

  • \(\bar{x} =\) sample mean
  • \(s_x =\) sample standard deviation (based on division by \(n - 1\))
  • \(\sigma_x =\) population standard deviation (based on division by \(n\))
  • minX \(=\) minimum
  • maxX \(=\) maximum
  • Q\(_1 =\) first Quartile
  • Med \(=\) median (or second Quartile)
  • Q\(_3 =\) third Quartile

The variance is equal to the standard deviation squared; Inter Quartile Range (IQR) \(=\) Q\(_3\) \(–\) Q\(_1\).

D.1.2 Binomial Distribution

Choose: 2nd VARS (= DISTR)

Let \(y \sim \mbox{Bin}(n,\ \pi)\) or \(y \sim \mbox{B}(n,\ \pi)\).

  • \(P(y = k) = \binom{n}{k} \times \pi^k \times (1 - \pi)^{n - k} = \frac{n!}{k! \times (n-k)!} \times \pi^k \times (1 - \pi)^{n - k}\): binompdf(\(n\),\(\pi\),\(k\))

  • \(P(y \leq k)\): binomcdf(\(n\),\(\pi\),\(k\))

Where \(n\) denotes the sample size, \(\pi\) denotes the population proportion of success, and \(k\) denotes the number of successes counted in the sample of size \(n\).

D.1.3 Normal Distribution

Choose: 2nd VARS (= DISTR)

Let \(y \sim \mbox{N}(\mu,\ \sigma)\).

  • \(P(\mbox{"lowerbound"} \leq y \leq \mbox{"upperbound"})\): normalcdf(lower, upper,\(\mu\),\(\sigma\))

  • \(P(y \geq \mbox{"lowerbound"})\): normalcdf(lower, 1E99,\(\mu\),\(\sigma\))

  • \(P(y \leq \mbox{"upperbound"})\): normalcdf(-1E99, upper,\(\mu\),\(\sigma\))

ImportantRemarks about calculating probabilities using a Normal Distributtion.

  • -1E99, being \(-1 \times 10^{99}\), denotes negative infinity (\(-\infty\)), and 1E99 (\(1 \times 10^{99}\)) denotes positive infinity (\(\infty\))

  • Use \(\mu = 0\) and \(\sigma = 1\) for the Standard Normal Distribution.

D.1.4 Inverse Normal Distribution

Choose: 2nd VARS (= DISTR)

Let \(y \sim \mbox{N}(\mu,\ \sigma)\).

  • invNorm(area,\(\mu\),\(\sigma\)) calculates the value \(q\) of \(y\) for which: \(P(y \leq q) =\) area

Therefore, area denotes a left-tailed probability under the normal distribution.

ImportantRemarks about calculating a quantile using the inverse Normal Distributtion.
  • Use \(\mu = 0\) and \(\sigma = 1\) for the Standard Normal Distribution.

D.1.5 Student t-distribution

Choose: 2nd VARS (= DISTR)

Let \(t \sim\) Student \(t\)-distribution with \(\nu =\) df degrees of freedom.

  • \(P(\mbox{"lowerbound"} \leq t \leq \mbox{"upperbound"})\): tcdf(lower, upper, df)

  • \(P(t \geq \mbox{"lowerbound"})\): tcdf(lower, 1E99, df)

  • \(P(t \leq \mbox{"upperbound"})\): tcdf(-1E99, upper, df)

ImportantRemarks about calculating probabilities using a Student t-distributtion.
  • -1E99, being \(-1 \times 10^{99}\), denotes negative infinity (\(-\infty\)), and 1E99 (\(1 \times 10^{99}\)) denotes positive infinity (\(\infty\))

D.1.6 Inverse Student t-distribution

Choose: 2nd VARS (= DISTR)

Let \(t \sim\) Student \(t\)-distribution with \(\nu =\) df degrees of freedom.

  • invT(area, df) calculates the value \(q\) of \(t\) for which: \(P(t \leq q) =\) area under a Student t-distribution with \(\nu =\) df degrees of freedom.

Therefore, area denotes a left-tailed probability under the Student t-distribution.

The invT() function is by default present in the TI-84 Plus graphing calculator. However, not by default in the TI-83/TI-83 Plus. Fortunately, this function can be easily added by programming it into the TI-83/TI-83 Plus as explained in the next section.

D.1.6.1 Programming the INVT() function into the TI-83/TI-83 Plus

Steps to program the INVT() function into the TI-83/TI-83 Plus graphing calculator, are shown in https://youtu.be/5Ft5eZVJtPk.

To execute the program select:

  1. PRGM, EXEC, got to INVT, and press ENTER to select.
  2. Press ENTER again and fill behind AREA LEFT: the value for the left-tailed probability.
  3. Press ENTER and fill behind DF: the value for the degrees of freedom \(\nu =\) df.
  4. After pressing ENTER the value for INVT(area, df) will be calculated as explained above.

D.2 Graphing Calculator Casio CFX-9850/fx-9750GII/fx-9860G Series

D.2.1 Measures of central tendency and variability

Choose: MENU > STAT or press 2 on the keypad, if necessary, empty existing lists via F\(\mathbf{6}\) > F\(\mathbf{4}\) DEL-A.

  • Type the different outcomes in List 1 and the corresponding frequencies in List 2.

  • Choose F\(\mathbf{2}\) CALC, and then F\(\mathbf{6}\) SET for setting the lists to use when needed:

    • 1 Var Xlist : List1
    • 1 Var Freq: List2 and press EXE

  • Select F\(\mathbf{1}\) 1VAR.

  • The following values, among others, are displayed:

    • \(\bar{x} =\) sample mean
    • \(s_x =\) sample standard deviation (based on division by \(n - 1\))
    • \(\sigma_x =\) population standard deviation (based on division by \(n\))
    • minX \(=\) minimum
    • Q1 \(=\) first Quartile
    • Med \(=\) median
    • Q3 \(=\) third quartile
    • maxX \(=\) maximum

The variance is equal to the standard deviation squared; Inter Quartile Range (IQR) \(=\) Q3 \(–\) Q1.

D.2.2 Binomial Distribution

Choose: MENU > STAT or press 2 on the keypad > F\(\mathbf{5}\) DIST

Let \(y \sim \mbox{Bin}(n,\ \pi)\) or \(y \sim \mbox{B}(n,\ \pi)\).

  • \(P(y = k) = \binom{n}{k} \times \pi^k \times (1 - \pi)^{n - k} = \frac{n!}{k! \times (n-k)!} \times \pi^k \times (1 - \pi)^{n - k}\):

    • Press F\(\mathbf{5}\) BINM and then F\(\mathbf{1}\) Bpd.
    • Press F\(\mathbf{2}\) Var to switch form list (F\(\mathbf{1}\) List) to variable mode, when needed.
    • Behind \(x\), enter the value for the number of successes \(k\) and press EXE.
    • Behind Numtrial, enter the value the number of trials \(n\) and press EXE.
    • Behind p, enter the value for the probability of success \(\pi\) and press EXE.
    • Navigate to Execute and select F\(\mathbf{1}\) CALC, or press EXE to calculate the answer.

  • \(P(y \leq k)\):

    • Press F\(\mathbf{5}\) BINM and then F\(\mathbf{2}\) Bcd.
    • Press F\(\mathbf{2}\) Var to switch form list (F\(\mathbf{1}\) List) to variable mode, when needed.
    • Behind \(x\), enter the value for the number of successes \(k\) and press EXE.
    • Behind Numtrial, enter the value the number of trials \(n\) and press EXE.
    • Behind p, enter the value for the probability of success \(\pi\) and press EXE.
    • Navigate to Execute and select F\(\mathbf{1}\) CALC, or press EXE to calculate the answer.

D.2.3 Normal Distribution

Choose: MENU > STAT or press 2 on the keypad > F\(\mathbf{5}\) DIST

Let \(y \sim \mbox{N}(\mu,\ \sigma)\).

  • Calculation of \(P(\mbox{"lowerbound"} \leq y \leq \mbox{"upperbound"}),\ P(y \geq \mbox{"lowerbound"}),\ \mbox{or } P(y \leq \mbox{"upperbound"})\):

    • Press F\(\mathbf{1}\) NORM, and then F\(\mathbf{2}\) Ncd.
    • Press F\(\mathbf{2}\) Var to switch form list (F\(\mathbf{1}\) List) to variable mode, when needed.
    • Behind Lower, type the “lowerbound” of the interval and press EXE. If there is no lowerbound use -1E99 (\(-1 \times 10^{99}\)) for negative infinity (\(-\infty\))
    • Behind Upper, type the “upperbound” of the interval and press EXE. If there is no upperbound use 1E99 (\(1 \times 10^{99}\)) for positive infinity (\(\infty\))
    • Provide the values for \(\sigma\), and \(\mu\).
    • Navigate to Execute and select F\(\mathbf{1}\) CALC, or press EXE to calculate the answer.
ImportantRemarks about calculating probabilities using a Normal Distributtion.
  • Use \(\mu = 0\) and \(\sigma = 1\) for the Standard Normal Distribution.

D.2.4 Inverse Normal Distribution

Choose: MENU > STAT or press 2 on the keypad > F\(\mathbf{5}\) DIST

Let \(y \sim \mbox{N}(\mu,\ \sigma)\).

  • Calculation of the value \(q\) of y for which: \(P(y \leq q) =\) Area:

    • Press F\(\mathbf{1}\) NORM and then F\(\mathbf{3}\) InvN.

    • Press F\(\mathbf{2}\) Var to switch form list (F\(\mathbf{1}\) List) to variable mode, when needed.

    • For the fx-9750GII/fx-9680G Series, please pay special attention to the side (Tail) used:

      • When F\(\mathbf{1}\) LEFT is used, the area from negative infinity (\(-\infty\)) to the upperbound \(q\) will be used, that is the left-tailed probability under the normal distribution \(P(y \leq q) =\) Area.
      • When F\(\mathbf{2}\) RIGHT is used, the area from the lowerbound \(q\) to positive infinity (\(\infty\)) will be used, that is the right-tailed probability under the normal distribution \(P(y \geq q) =\) Area.
      • When F\(\mathbf{3}\) CNTR is used, the area from the lowerbound \(-q\) to the upperbound \(q\) will be used, that is the two-tailed probability under the normal distribution \(P(-q \leq y \leq q) =\) Area.

    • Behind Area, fill the value for the probability under the normal distribution. In case of using a Casio CFX-9850 Series enter the left-tailed probability under the normal distribution, being \(P(y \leq q) =\) Area.

    • Provide the values for \(\sigma\), and \(\mu\).

    • Navigate to Execute and select F\(\mathbf{1}\) CALC, or press EXE to calculate the answer.

ImportantRemarks about calculating probabilities using a Normal Distributtion.
  • Use \(\mu = 0\) and \(\sigma = 1\) for the Standard Normal Distribution.

D.2.5 Student t-distribution

Choose: MENU > STAT or press 2 on the keypad > F\(\mathbf{5}\) DIST

Let \(t \sim\) Student \(t\)-distribution with \(\nu =\) df degrees of freedom.

  • Calculation of \(P(\mbox{"lowerbound"} \leq y \leq \mbox{"upperbound"}),\ P(y \geq \mbox{"lowerbound"}),\ \mbox{or } P(y \leq \mbox{"upperbound"})\):

    • Press F\(\mathbf{2}\) t, and then F\(\mathbf{2}\) tcd.
    • Press F\(\mathbf{2}\) Var to switch form list (F\(\mathbf{1}\) List) to variable mode, when needed.
    • Behind Lower, type the “lowerbound” of the interval and press EXE. If there is no lowerbound use -1E99 (\(-1 \times 10^{99}\)) for negative infinity (\(-\infty\))
    • Behind Upper, type the “upperbound” of the interval. If there is no upperbound use 1E99 (\(1 \times 10^{99}\)) for positive infinity (\(\infty\))
    • Provide the values for \(\nu =\) df degrees of freedom.
    • Navigate to Execute and select F\(\mathbf{1}\) CALC, or press EXE to calculate the answer.

D.2.6 Inverse Student t-distribution

Choose: MENU > STAT or press 2 on the keypad > F\(\mathbf{5}\) DIST

Let \(t \sim\) Student \(t\)-distribution with \(\nu =\) df degrees of freedom.

  • Calculate the value \(q\) of \(t\) for which: \(P(t \leq q) =\) Area:

    • Press F\(\mathbf{2}\) t, and then F\(\mathbf{3}\) Invt.
    • Press F\(\mathbf{2}\) Var to switch form list (F\(\mathbf{1}\) List) to variable mode, when needed.
    • Behind Area, fill the value for the right-tailed probability under the Student t-distribution, that is \(P(y \geq q) =\) Area.
    • Provide the value for the degrees of freedom \(\nu =\) df.
    • Navigate to Execute and select F\(\mathbf{1}\) CALC, or press EXE to calculate the answer.