Trail Mix StatisticsBy Tyler Lipshutz

About Trail MixMade with dried fruits, grains, nuts,

and sometimes chocolateIdeal for long outdoor hikes1968 when first invented in CaliforniaGORP, “Good Old Raisins and Peanuts”PlantersEmeraldArcher FarmsThird Party brands (Wegmans)

My Goal:Is the distribution in serving sizes

equal to the entire bag? Does the bags overall weight distribute equal to the estimated amount of servings?

How I collected my dataTwo different trail mixes

◦Planters & Wegmans◦$1.59/ 6 oz. & $4.69/ 16 oz.

Count the amount in each bagFind the weight of the servingPick one random piece and count

how many were in that serving.Record data into charts

Distribution of Pieces – Planters.Peanuts (53%)Raisins(24%)M&Ms(17%)Almonds(3%)Cashews(3%) Peanuts

53%

M&Ms17%

Cashew s3%

Raisins24%

Almonds3%

Peanuts M&Ms Cashew s Raisins Almonds

Distribution of Pieces – Rainbow DelightPeanuts (59%)Raisins(14%)M&Ms(21%)Almonds(3%)Cashews(3%)

Peanuts M&Ms Cashews Raisins Almonds

Brand vs. WeightShape:

◦ Both are roughly unimodal, P has two peaks around 34 g and 46 g with a gap at 36. RD has peaks at 32 g and 42 g and a gap at 43.

Center: Planters has a center at 34. 9 g. RD has a center at 36.0323 g.

Spread: Planters has a wider spread with a range of 41 and RD has a range of 27.

Rainbow Delight Serving Weight (g)

Weight_g_RD

36.0323213135414831

6.61052S1 = meanS2 = minS3 = Q1S4 = medianS5 = Q3S6 = maxS7 = countS8 = stdDev

Planters Serving Size Weight (g)

Weight_g_P

34.9122834435330

9.23767S1 = meanS2 = minS3 = Q1S4 = medianS5 = Q3S6 = maxS7 = countS8 = stdDev

Product

0

1

2

3

4

5

6

P

1

2

3

4

5

6

RD

Weight_g0 10 20 30 40 50 60

Trail Mix Histogram

Quantitative Data – Product vs. PeanutsShape: Planters is right

skewed and RD is close to unimodal

Center: Planters mean is 17.2 and RD is 21.1667

Spread: Planters has the greater spread with a range of 17 but Rainbow Delight has a range of 14. Planters has a stdDev of 7.4632 and Rainbow Delight has a stdDev of 5.1929.

PR

DPeanuts

0 5 10 15 20 25 30

Trail Mix Box Plot

Trail Mix

Column Summary

Peanuts

P

RD

Product

517.2

1112132228

621.1667

1516

21.52429

1119.3636

1113212429

S1 = countS2 = meanS3 = minS4 = Q1S5 = medianS6 = Q3S7 = max

Quantitative Data – Product vs. M&MsShape: Both are slightly

unimodalCenter: Planters mean is

4.33 and RD is 7.625Spread: Both have the same

range of 9 so they are equally wide. Planters has a stdDev of 3 and Rainbow Delight has a stdDev of 2.92.

Overall both have an equal spread but Planters has a much lower amount of M&Ms per serving size than Wegman’s Brand

PR

D

MMs0 2 4 6 8 10 12 14

Trail Mix Box Plot

Trail Mix

Column Summary

MMs

P

RD

Product

94.33333

034693

87.625

45.57.5

913

2.92465

175.88235

0468

133.33321

S1 = countS2 = meanS3 = minS4 = Q1S5 = medianS6 = Q3S7 = maxS8 = s

Quantitative Data – Product vs. Cashews Shape: Planters has a right skew

but the shape of RD looks unimodal Center: Planters mean is 1.25 and

RD is 2.714 Spread: Rainbow Delight has a

higher range of 6 which means a wider spread when compared to Planters with has a range of 4. Planters has a stdDev of 1.8929 and Rainbow Delight has a stdDev of 2.058.

Overall, both have little average amounts of cashews in them when compared to other pieces but of these two brands Wegman’s brand has a larger average cashews.

Trail Mix

Column Summary

Cashews

P

RD

Product

41.25

00

0.52.5

41.89297

72.71429

01246

2.05866

112.18182

00246

2.0405S1 = countS2 = meanS3 = minS4 = Q1S5 = medianS6 = Q3S7 = maxS8 = s

PR

DCashews

0 1 2 3 4 5 6 7

Trail Mix Box Plot

Quantitative Data – Product vs. Raisins Shape: Both brands look

unimodal for Raisins Center: Planters mean is 9.333

and RD is 18.5 Spread: Rainbow Delight has a

smaller range of 3 but Planters is much wider with 9. Planters has a stdDev of 3.559 and Rainbow Delight has a stdDev of 1.29.

Overall, Rainbow Delight’s average is higher than Planters per serving but there only 4 samples from Rainbow delight so the data might not be justified.

PR

D

Raisins0 2 4 6 8 10 12 14 16 18 20 22

Trail Mix Box Plot

Trail Mix

Column Summary

Raisins

P

RD

Product

69.33333

56

9.51214

3.55903

418.5

1717.518.519.5

201.29099

101358

131820

5.47723S1 = countS2 = meanS3 = minS4 = Q1S5 = medianS6 = Q3S7 = maxS8 = s

Quantitative Data – Product vs. Almonds Shape: Planters is right

skewed but RD looks unimodal Center: Planters mean is

2.6667 and RD is 4.8333 Spread: Rainbow Delight has

a larger range of 9 but Planters is smaller with 5. Planters has a stdDev of 1.751 and Rainbow Delight has a stdDev of 3.430.

Overall, Rainbow Delight’s average is higher than Planters per serving and RD’s mean matches Planter’s max and the Q1 for both are the same at 2.

PR

D

Almonds0 2 4 6 8 10 12

Trail Mix Box Plot

Trail Mix

Column Summary

Almonds

P

RD

Product

62.66667

02

2.545

1.75119

64.83333

12

4.57

103.43026

123.75

023

5.510

2.83244S1 = countS2 = meanS3 = minS4 = Q1S5 = medianS6 = Q3S7 = maxS8 = s

1 Samp T Interval Conditions: SRSPop>10nNormal Pop

or > 30

Random weighing

There are more than 300 servings of trail mix; there is more than 310 servings of trail mix

30>30; 31>30Conditions Med >> Students’ t-model >> 1 Samp T Interval

1 Sample T Interval: Rainbow Delight weight per serving

N = 31, df = 30Sample mean = 36.032336.0323 + 2.424We are 95% confident that the true

mean of grams per serving of Wegmans Trail Mix from Wegmans is between 33.6075 and 38.457

1 Sample T Interval: Planters Trail Mix weight per serving

N = 30, df = 29Sample mean = 34.934.9 + 3.449We are 95% confident that the true

mean of grams per serving of Planters Trail Mix from Wegmans is between 31.451 and 38.451

Chi-Squared GOF ConditionsCategorical

Data?SRS?All expected cell

counts > 5?

Yes, bothStated, bothYes all expected

for both > 5

(218-76.4)²+(60-76.4)²+…=398.49476.44 76.4 76.4

P( >398.4947|df=4)= 5.88x10-85

Ho: The observed frequency distribution of pieces fits the expected distribution.

We reject the Ho because our p-value is < alpha which is .05. We have sufficient evidence that the observed frequency distribution of pieces does not fit the expected distribution.

Chi-Squared GOF Test - Planters

Ha: The observed frequency distribution of pieces does not fit the expected distribution

Peanuts M&Ms Cashews Raisins AlmondsPieces/Bag 218(76.4) 60(76.4) 6(76.4) 91(76.4) 6(76.4)

(636-217.1)²+(226.5-217.1)²+…=1131.527867 217.1 217.1

P( >1131.527867|df=4)= 0

Ho: The observed frequency distribution of pieces fits the expected distribution.

We reject the Ho because our p-value is < alpha which is .05. We have sufficient evidence that the observed frequency distribution of pieces does not fit the expected distribution.

Chi-Squared GOF Test – Rainbow Delight

Ha: The observed frequency distribution of pieces does not fit the expected distribution

Peanuts M&Ms Cashews Raisins Almonds

Pieces/Bag 636(217.1)226.5(217.

1) 38(217.1)151.5(217.

1)33.5(217.1

)

Bias & ErrorSome peanuts were still together

while others were split in half◦ I counted both as one(1).

Raisins were sometimes conjoined

While counting one would fall on the floor

Human error of counting/Loss of Precision

I got hungry !

Personal Opinions & ConclusionPlanters is tastierMore convenient than a giant tubBased of the GOF tests, neither have

uniformly distributed pieces.The Rainbow Delight has an accurate

serving size based on the confidence interval and the claim on the tub.

The serving size for Planters is higher based on the confidence interval compared to the claim on the bag.