Entomology Worksheet.docx - Entomology Worksheet Student Name: Ben ... - Free Printable

Let’s go step by step to solve this problem.

We are given a table with data about students’ heights and weights, and we’re asked to find the correlation coefficient between height (in inches) and weight (in pounds). The formula for the correlation coefficient \( r \) is:

\[
r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}}
\]

Where:
- \( n \) = number of pairs (students)
- \( x \) = height
- \( y \) = weight
- \( \sum xy \) = sum of each height multiplied by its corresponding weight
- \( \sum x \), \( \sum y \) = sums of all heights and weights
- \( \sum x^2 \), \( \sum y^2 \) = sums of squares of heights and weights

---

From Table 1 in the image (we’ll extract the values):

Student | Height (x) | Weight (y)
--------|------------|----------
1 | 60 | 95
2 | 63 | 105
3 | 65 | 110
4 | 67 | 120
5 | 70 | 130

So, \( n = 5 \)

Now let’s compute each part:

Step 1: Compute Σx, Σy, Σxy, Σx², Σy²

Σx = 60 + 63 + 65 + 67 + 70
= 60+63=123; 123+65=188; 188+67=255; 255+70=325

Σy = 95 + 105 + 110 + 120 + 130
= 95+105=200; 200+110=310; 310+120=430; 430+130=560

Σxy = (60×95) + (63×105) + (65×110) + (67×120) + (70×130)

Calculate each:

60 × 95 = 5700
63 × 105 = 6615
65 × 110 = 7150
67 × 120 = 8040
70 × 130 = 9100

Add them:
5700 + 6615 = 12315
12315 + 7150 = 19465
19465 + 8040 = 27505
27505 + 9100 = 36605

Σx² = 60² + 63² + 65² + 67² + 70²
= 3600 + 3969 + 4225 + 4489 + 4900

Compute:

3600 + 3969 = 7569
7569 + 4225 = 11794
11794 + 4489 = 16283
16283 + 4900 = 21183

Σy² = 95² + 105² + 110² + 120² + 130²
= 9025 + 11025 + 12100 + 14400 + 16900

Compute:

9025 + 11025 = 20050
20050 + 12100 = 32150
32150 + 14400 = 46550
46550 + 16900 = 63450

---

Step 2: Plug into the formula

Numerator:
\( n(\sum xy) - (\sum x)(\sum y) = 5(36605) - (325)(560) \)

5 × 36605 = 183025
325 × 560 → Let’s compute:
300×560 = 168000
25×560 = 14000
Total = 168000 + 14000 = 182000

So numerator = 183025 - 182000 = 1025

Denominator:
First part: \( n\sum x^2 - (\sum x)^2 = 5(21183) - (325)^2 \)

5 × 21183 = 105915
325² = 325 × 325 → 300²=90000, 25²=625, 2×300×25=15000 → total = 90000+15000+625=105625

So first part = 105915 - 105625 = 290

Second part: \( n\sum y^2 - (\sum y)^2 = 5(63450) - (560)^2 \)

5 × 63450 = 317250
560² = 313600

So second part = 317250 - 313600 = 3650

Now denominator = √(290 × 3650)

Compute 290 × 3650:

Break it down:
290 × 3650 = 290 × (3000 + 650) = 290×3000 + 290×650

290×3000 = 870,000
290×650 = 290×600 + 290×50 = 174,000 + 14,500 = 188,500
Total = 870,000 + 188,500 = 1,058,500

So denominator = √1,058,500

Let’s approximate √1,058,500:

Note that 1029² = ?
1000² = 1,000,000
1030² = (1000+30)² = 1000000 + 2×1000×30 + 900 = 1,000,000 + 60,000 + 900 = 1,060,900 → too big

Try 1028²:
(1030 - 2)² = 1030² - 2×1030×2 + 4 = 1,060,900 - 4,120 + 4 = 1,056,784

Our value is 1,058,500 — between 1028² and 1029²

1029² = 1028² + 2×1028 + 1 = 1,056,784 + 2056 + 1 = 1,058,841 → very close!

1,058,841 vs our 1,058,500 → difference of 341

So √1,058,500 ≈ 1028.8 (since 1029² is 1,058,841, which is 341 over, so subtract roughly 341/(2×1029) ≈ 341/2058 ≈ 0.166 → so approx 1028.83)

But actually, let’s use calculator-style precision if allowed — but since this is homework, maybe we can leave as exact or round later.

Wait — perhaps I made an error? Let me double-check multiplication:

Actually, 290 × 3650:

Alternative way:
290 × 3650 = 29 × 10 × 365 × 10 = 29 × 365 × 100

29 × 365:

30×365 = 10,950 minus 1×365 = 365 → 10,950 - 365 = 10,585

Then ×100 = 1,058,500 → correct.

Now √1,058,500 — let's see if it simplifies.

Factor 1,058,500:

Divide by 100: 10585 × 100 → √(10585 × 100) = 10√10585

Check if 10585 has square factors:

10585 ÷ 5 = 2117
2117 ÷ 29 = 73 (because 29×73 = 2117? 30×73=2190, minus 73=2117 → yes!)

So 10585 = 5 × 29 × 73 → no repeated primes → so √10585 doesn’t simplify.

Thus, √1,058,500 = 10√10585 ≈ 10 × 102.88 ≈ 1028.8 (as before)

But wait — let’s check actual calculation again because 1029² = 1,058,841, and we have 1,058,500 — difference 341.

So √1,058,500 ≈ 1029 - (341)/(2×1029) ≈ 1029 - 341/2058 ≈ 1029 - 0.1657 ≈ 1028.834

So denominator ≈ 1028.834

Numerator = 1025

So r ≈ 1025 / 1028.834 ≈ ?

1025 ÷ 1028.834 ≈ 0.9963

That seems extremely high — almost perfect correlation. Is that right?

Let me verify the numbers again — maybe I miscalculated Σxy or something.

Recheck Σxy:

Student 1: 60×95 = 5700 ✔️
Student 2: 63×105: 60×105=6300, 3×105=315 → 6615 ✔️
Student 3: 65×110 = 7150 ✔️
Student 4: 67×120: 60×120=7200, 7×120=840 → 8040 ✔️
Student 5: 70×130=9100 ✔️
Sum: 5700+6615=12315; +7150=19465; +8040=27505; +9100=36605 ✔️

Σx=325, Σy=560 → product=325×560

325×500=162500, 325×60=19500 → total 182000 ✔️

5×36605=183025 → 183025-182000=1025 ✔️

Σx²: 60²=3600, 63²=3969, 65²=4225, 67²=4489, 70²=4900

3600+3969=7569; +4225=11794; +4489=16283; +4900=21183 ✔️

5×21183=105915; 325²=105625; diff=290 ✔️

Σy²: 95²=9025, 105²=11025, 110²=12100, 120²=14400, 130²=16900

9025+11025=20050; +12100=32150; +14400=46550; +16900=63450 ✔️

5×63450=317250; 560²=313600; diff=3650 ✔️

290×3650=1,058,500 ✔️

√1,058,500 ≈ 1028.834

1025 / 1028.834 ≈ 0.9963

But let’s compute more precisely:

1025 ÷ 1028.834 = ?

Do division: 1025 / 1028.834 = ?

Or better: use fractions.

r = 1025 / √(290 × 3650)

But 290 × 3650 = 1,058,500

Is there a common factor?

Notice: 1025 and 1,058,500 — both divisible by 25?

1025 ÷ 25 = 41
1,058,500 ÷ 25 = 42,340

Not helpful.

Alternatively, compute numerically:

√1,058,500 = sqrt(1058500)

Use calculator approximation: sqrt(1058500) = 1028.834... as above.

1025 / 1028.834 ≈ 0.99627

So r ≈ 0.996

But typically, we report correlation coefficients to 3 decimal places.

However, let me check if the problem expects us to use a different method or if I missed something.

Looking back at the task: “using the given data, find the correlation coefficient”

And the table is small — only 5 points.

Perhaps they want us to use the computational formula correctly.

Another way: sometimes people use means and standard deviations, but the formula given is the direct one.

Maybe I should compute using another approach to verify.

Let me compute mean of x and y.

Mean x = 325 / 5 = 65
Mean y = 560 / 5 = 112

Now compute covariance and standard deviations.

Covariance s_xy = [Σ(xy) - n * mean_x * mean_y] / (n-1) ? But for population correlation, sometimes divided by n.

The formula we used is for sample correlation coefficient, which divides by n in the sums but uses n in the formula — actually, the formula we used is standard for Pearson r, and it’s the same whether you consider sample or population for this context.

In many textbooks, for such problems, they use the formula without dividing by n-1 — it’s built into the formula.

Our calculation seems consistent.

But 0.996 is very high — let’s plot mentally:

Heights: 60,63,65,67,70 — increasing
Weights: 95,105,110,120,130 — also increasing steadily

Differences:

From 60 to 63 (+3), weight +10
63 to 65 (+2), weight +5
65 to 67 (+2), weight +10
67 to 70 (+3), weight +10

Not perfectly linear, but very strong positive trend.

Let me compute the actual line of best fit slope to see.

Slope b = [nΣxy - ΣxΣy] / [nΣx² - (Σx)²] = 1025 / 290 ≈ 3.5345

Then intercept a = mean_y - b*mean_x = 112 - 3.5345*65 ≈ 112 - 229.7425 ≈ -117.7425

Now predict weights:

For x=60: y_pred = 3.5345*60 -117.7425 ≈ 212.07 -117.7425 ≈ 94.3275 → actual 95 → close
x=63: 3.5345*63≈222.6735 -117.7425≈104.931 → actual 105 → very close
x=65: 3.5345*65≈229.7425 -117.7425=112 → actual 110 → off by 2
x=67: 3.5345*67≈236.8115 -117.7425≈119.069 → actual 120 → close
x=70: 3.5345*70≈247.415 -117.7425≈129.6725 → actual 130 → very close

Residuals are small, so correlation should be very high.

Compute R-squared from regression: but anyway, our r=0.996 seems plausible.

But let me calculate exactly:

r = 1025 / sqrt(290 * 3650)

Compute 290 * 3650 = 1,058,500

sqrt(1,058,500) = ?

Let me calculate more accurately.

1028.834^2 = ? Earlier we had 1029^2=1,058,841

1028.8^2 = (1029 - 0.2)^2 = 1029^2 - 2*1029*0.2 + 0.04 = 1,058,841 - 411.6 + 0.04 = 1,058,429.44

Our target is 1,058,500 — higher than 1,058,429.44

Difference: 1,058,500 - 1,058,429.44 = 70.56

Increment by dx: derivative of x^2 is 2x, so dx = 70.56 / (2*1028.8) ≈ 70.56 / 2057.6 ≈ 0.0343

So sqrt ≈ 1028.8 + 0.0343 ≈ 1028.8343 — same as before.

So r = 1025 / 1028.8343 ≈ ?

Compute 1025 ÷ 1028.8343

= 1 - (3.8343)/1028.8343 ≈ 1 - 0.003726 ≈ 0.996274

So r ≈ 0.996

Typically rounded to three decimals: 0.996

But let me confirm with exact fraction.

r = 1025 / sqrt(290 * 3650)

Simplify inside: 290 = 29*10, 3650=365*10=73*5*10, so 290*3650=29*10*73*5*10=29*73*5*100

So sqrt(29*73*5*100) = 10 * sqrt(29*73*5) = 10*sqrt(10585)

As before.

10585 = 5*29*73 — no squares, so irrational.

Numerically, r = 1025 / (10 * sqrt(10585)) = 102.5 / sqrt(10585)

sqrt(10585) ≈ 102.8834 (since 102.8834^2 = ? 100^2=10000, 2.8834^2≈8.31, 2*100*2.8834=576.68, total≈10000+576.68+8.31=10584.99 — close)

So 102.5 / 102.8834 ≈ 0.99627

Same thing.

Therefore, the correlation coefficient is approximately 0.996

But in some contexts, they might expect more precision or rounding.

Looking at the data, perhaps they want it to three decimal places.

I think 0.996 is fine.

But let me see if I can get an exact value or if I made a mistake in interpretation.

Another thought: in the formula, is it for sample or population? In most introductory stats, for such problems, they use this formula and it gives the sample correlation coefficient.

Sometimes they divide by n-1 in variance, but in this formula, it's already accounted for.

I recall that the formula we used is standard and correct.

Perhaps the answer is expected to be 0.996.

But let me calculate using software-like precision.

Assume:

n = 5
sum_x = 325
sum_y = 560
sum_xy = 36605
sum_x2 = 21183
sum_y2 = 63450

numerator = 5*36605 - 325*560 = 183025 - 182000 = 1025

denom_x = 5*21183 - 325^2 = 105915 - 105625 = 290

denom_y = 5*63450 - 560^2 = 317250 - 313600 = 3650

product = 290 * 3650 = 1,058,500

sqrt_product = sqrt(1058500) = let's say we use calculator: sqrt(1058500) = 1028.834292...

r = 1025 / 1028.834292 ≈ 0.996274

Rounded to three decimal places: 0.996

If rounded to four: 0.9963, but usually three is sufficient.

In the context of the problem, since it's a school task, likely 0.996 is acceptable.

But let me check if the problem has any instruction on rounding — not visible, but typically three decimals.

Perhaps they want the exact fraction, but unlikely.

Another idea: maybe I misread the table.

Looking back at the user's image description — but I don't have the image, only the text.

In the initial request, it says "Table 1" with students 1 to 5, heights and weights as I used.

Perhaps student 3 weight is 110, etc.

I think my calculation is correct.

So final answer should be 0.996

But to be thorough, let's compute r using another formula.

r = cov(x,y) / (s_x * s_y)

cov(x,y) = [Σ(xy) - n * mean_x * mean_y] / (n-1) for sample covariance? But for correlation, it cancels out.

Actually, the Pearson r is the same regardless.

Sample covariance s_xy = [Σ(xy) - (Σx Σy)/n] / (n-1)

But then s_x = sqrt[ (Σx² - (Σx)²/n ) / (n-1) ]

Similarly for s_y.

Then r = s_xy / (s_x s_y)

Let me try that to verify.

First, mean_x = 65, mean_y = 112

Σ(xy) = 36605

So Σ(xy) - n * mean_x * mean_y = 36605 - 5*65*112

65*112 = 65*100=6500, 65*12=780, total 7280

5*7280=36400

So 36605 - 36400 = 205

This is the numerator for covariance if divided by n, but for sample covariance, divide by n-1=4.

In the correlation formula, when we do r = [Σ(xy) - n mean_x mean_y] / sqrt[ (Σx² - n mean_x²) (Σy² - n mean_y²) ]

Which is exactly what we did earlier, because:

Σx² - n mean_x² = Σx² - (Σx)²/n

Similarly for y.

And in our case:

Σx² - (Σx)²/n = 21183 - (325)^2 / 5 = 21183 - 105625 / 5 = 21183 - 21125 = 58? Wait no.

Earlier we had n Σx² - (Σx)² = 5*21183 - 105625 = 105915 - 105625 = 290

But Σx² - (Σx)²/n = 21183 - 105625/5 = 21183 - 21125 = 58

Similarly, Σy² - (Σy)²/n = 63450 - 313600/5 = 63450 - 62720 = 730

And Σxy - (Σx Σy)/n = 36605 - 182000/5 = 36605 - 36400 = 205

Then r = 205 / sqrt(58 * 730)

Compute 58 * 730 = 58*700=40600, 58*30=1740, total 42340

sqrt(42340) ≈ ? 205.77 or something

205 / sqrt(42340)

But earlier we had r = 1025 / sqrt(1058500)

Note that 1025 / sqrt(1058500) = 1025 / sqrt(25 * 42340) = 1025 / (5 * sqrt(42340)) = 205 / sqrt(42340)

Same thing!

So r = 205 / sqrt(42340)

Now sqrt(42340): 205.77? 206^2=42436, 205^2=42025

42340 - 42025 = 315

dx = 315 / (2*205) ≈ 315/410 ≈ 0.768

so sqrt≈205.768

Then r = 205 / 205.768 ≈ 0.99627 — same as before.

So confirmed.

Therefore, the correlation coefficient is 0.996

Final Answer:
0.996