## Formula-Based Basic Statistics Questions in New Quizzes

Community Participant

Canvas has a limited set of functions it provides for use in formula questions. James has already written Blog posts with some work-arounds. Below, I give concrete examples of Canvas New Quizzes formula questions for hypothesis testing and confidence intervals expanding on James's ideas using known approximations to the normal distribution, inverse normal distribution, and t-distribution. Approximations are good only to about 2-3 decimal places.

Example of a standard normal distribution question:

Dinners at four star restaurants in a certain city cost on average $X with a standard deviation of$S and are normally distributed. You go out to eat at random at these restaurants n times. What is the probability that the average you spend is greater than \$50? Round your answer to two decimal places.

Variables:

 Variable Min Max Decimals S 10 20 0 X 44 49 0 n 10 20 0

Formula Definition:

a = (50 -X)/(S/sqrt(n))

cdf = if(a,if(a + abs(a),
1-(0.31938153/(1+0.2316419*a)-0.356563782/(1+0.2316419*a)^2+
1.781477937/(1+0.2316419*a)^3-1.821255978/(1+0.2316419*a)^4+
1.330274429/(1+0.2316419*a)^5)*e^(-0.5*a^2)/sqrt(2*pi),
(0.31938153/(1-0.2316419*a)-0.356563782/(1-0.2316419*a)^2+
1.781477937/(1-0.2316419*a)^3-1.821255978/(1-0.2316419*a)^4+
1.330274429/(1-0.2316419*a)^5)*e^(-0.5*a^2)/sqrt(2*pi)),
0.5)

1-cdf

Generate Possible Solutions:

Number of solutions: 200. Decimal Places: 4. Margin type: absolute. +/- margin of error: 0.005.

Example of an inverse standard normal distribution question:

Suppose that weights in pounds of bags of flour follow a normal distribution with a standard deviation of 0.5 lbs. If PP% of bags are heavier than x lbs, what is the mean of this distribution?  Round to two decimal places.

Variables:

 Variable Min Max Decimals PP 1 10 0 x 100 125 0

Formula Definition:

P = PP/100
R = if( (P - 0.5) + abs(P - 0.5), 1-P, P)

Y = sqrt(-2*ln(R))

Z= if((R - 0.5), if( (P - 0.5) + abs(P - 0.5), Y - ((((0.0000453642210148*Y + 0.0204231210245)*Y + 0.342242088547)*Y+1)*Y + 0.322232431088) / ((((0.0038560700634*Y + 0.10353775285)*Y+0.531103462366)*Y +0.588581570495)*Y + 0.099348462606),((((0.0000453642210148*Y + 0.0204231210245)*Y + 0.342242088547)*Y+1)*Y + 0.322232431088) / ((((0.0038560700634*Y + 0.10353775285)*Y+0.531103462366)*Y +0.588581570495)*Y + 0.099348462606) -Y ), 0)
x + 0.5*Z

Generate Possible Solutions:

Number of solutions: 200. Decimal Places: 6. Margin type: absolute. +/- margin of error: 0.005.

Example of a t-distribution question:

Find the probability that a T-distribution with nu degrees of freedom is less than z.  Round your answer to two decimal places.

Variables:

 Variable Min Max Decimals nu 5 25 0 z -3.00 3.00 2

Formula Definition:

T = abs(z)
tanh(x) = (e^(2*x) -1)/(e^(2*x) + 1)
x=-0.06148*nu + 0.011443*T + 0.841444
H14 = tanh(x)
x = -0.011259*nu - 0.31433*T + 1.125778
H13 =tanh(x)
x = 0.219778*nu - 0.212474*T + 0.776667
H12 = tanh(x)
x = 0.015481*nu - 0.268557*T + 0.22856
H11 = tanh(x)
x = -0.726 + 1.769*H11 - 1.304*H12 + 0.284*H13 + 0.78*H14
H24 = tanh(x)
x = -1.545 + 1.796*H11- 0.115*H12 + 0.918*H13 + 0.701*H14
H23 = tanh(x)
x = 0.936 - 0.613*H11 + 1.339*H12 - 1.148*H13 - 0.796*H14
H22 = tanh(x)
x = -2.013 + 1.718*H11 - 0.043*H12 + 1.346*H13 + 0.39*H14
H21 = tanh(x)
y = 0.259 - 1.435*H21 + 0.604*H22 + 0.548*H23 + 0.75*H24
if(z, if(z + abs(z), y, 1-y), 0.5)

Generate Possible Solutions:

Number of solutions: 200. Decimal Places: 4. Margin type: absolute. +/- margin of error: 0.005.

These formulas came from James's blog post, the Odeh and Evans formula in
https://link.springer.com/content/pdf/10.3758/BF03200956.pdf, and the approximation for the T Distribution in  https://statperson.com/Journal/StatisticsAndMathematics/Article/Volume8Issue1/IJSAM_8_1_4.pdf

Drawing on James's list idea for a class that uses Excel for the exams, the following uses the list feature to "look up" the relevant t-distribution value.

Example of a one-sample confidence interval question:

Find the lower limit of a PP% confidence interval for the mean number of calories in a McDonald's Big Mac given the sample values from 10 random Big Macs provided in the table below. Round your answer to two decimal places.

 Calories in the sampled Big Mac x1 x2 x3 x4 x5 x6 x7 x8

Variables:

 Variable Min Max Decimals PP 96 99 0 x1 575 650 0 x2 575 650 0 x3 575 650 0 x4 575 650 0 x5 575 650 0 x6 575 650 0 x7 575 650 0 x8 575 650 0

Formula Definition:

xbar = mean(x1, x2, x3, x4, x5, x6, x7, x8)
sumsq= sum((x1 - xbar)^2, (x2 - xbar)^2, (x3 - xbar)^2, (x4 - xbar)^2, (x5 - xbar)^2, (x6 - xbar)^2, (x7 - xbar)^2, (x8 - xbar)^2)
s = sqrt(sumsq/7)
tarray = reverse(reverse(3.499483297, 2.997951567, 2.714573011, 2.516752424))
entry = (99-PP)
xbar - (s/sqrt(8))*at(tarray, entry)

Generate Possible Solutions:

Number of solutions: 200. Decimal Places: 3. Margin type: absolute. +/- margin of error: 0.005.