Question

7 a doctor recorded the weight of five of his patients. the weight of each patient is: 29, 20, 21, 19, 35, 28, 33, 31. complete the chart to: determine the mean weight; determine the difference of each patient from the mean weight; square each differences; find the sum of the squared differences. determine the variance of the patients weights and the sample standard deviation. round the standard deviation to the nearest hundredth. data set 1 x x̄ x - x̄ (x - x̄)² σ(x - x̄)² / n standard deviation: √σ(x - x̄)² / n

Explanation:

Step1: Calculate the mean ($\bar{x}$)

First, find the sum of the weights. The weights are 29, 20, 21, 19, 35, 28, 33, 31. Wait, wait, the problem says five patients but the data has 8 values? Wait, maybe a typo, but let's check: 29,20,21,19,35,28,33,31 – that's 8 values. Let's sum them: 29 + 20 = 49; 49 +21=70; 70+19=89; 89+35=124; 124+28=152; 152+33=185; 185+31=216. Wait, no, 29+20=49, +21=70, +19=89, +35=124, +28=152, +33=185, +31=216? Wait 29+20=49, +21=70, +19=89, +35=124, +28=152, +33=185, +31=216? Wait 29+20=49, 49+21=70, 70+19=89, 89+35=124, 124+28=152, 152+33=185, 185+31=216. Wait, but the number of values n: let's count the data points: 29,20,21,19,35,28,33,31 – 8 values. So n=8. Then mean $\bar{x} = \frac{\sum x}{n} = \frac{29 + 20 + 21 + 19 + 35 + 28 + 33 + 31}{8}$. Let's recalculate the sum: 29+20=49, +21=70, +19=89, +35=124, +28=152, +33=185, +31=216. Wait 29+20=49, 49+21=70, 70+19=89, 89+35=124, 124+28=152, 152+33=185, 185+31=216. So sum is 216? Wait no, 29+31=60, 20+33=53, 21+28=49, 19+35=54. Then 60+53=113, +49=162, +54=216. Yes, sum is 216. Then mean $\bar{x} = \frac{216}{8} = 27$.

Step2: Calculate $x - \bar{x}$ and $(x - \bar{x})^2$ for each x

For each x:
- x=29: $29 - 27 = 2$; $(2)^2 = 4$
- x=20: $20 - 27 = -7$; $(-7)^2 = 49$
- x=21: $21 - 27 = -6$; $(-6)^2 = 36$
- x=19: $19 - 27 = -8$; $(-8)^2 = 64$
- x=35: $35 - 27 = 8$; $(8)^2 = 64$
- x=28: $28 - 27 = 1$; $(1)^2 = 1$
- x=33: $33 - 27 = 6$; $(6)^2 = 36$
- x=31: $31 - 27 = 4$; $(4)^2 = 16$

Step3: Sum of squared differences ($\sum (x - \bar{x})^2$)

Sum the $(x - \bar{x})^2$ values: 4 + 49 + 36 + 64 + 64 + 1 + 36 + 16. Let's calculate: 4+49=53; 53+36=89; 89+64=153; 153+64=217; 217+1=218; 218+36=254; 254+16=270. So $\sum (x - \bar{x})^2 = 270$.

Step4: Calculate variance (population variance, since it's all patients? Wait the problem says "variance of the patients' weights" – if it's the entire population (n=8), variance $\sigma^2 = \frac{\sum (x - \bar{x})^2}{n} = \frac{270}{8} = 33.75$. But if it's sample variance, it would be $\frac{270}{7}$, but the problem says "variance" and "sample standard deviation"? Wait the chart has $\frac{\sum (x - \bar{x})^2}{n}$ for variance? Wait the chart's first box under variance is $\frac{\sum (x - \bar{x})^2}{n}$, so maybe population variance. Then sample standard deviation would be $\sqrt{\frac{\sum (x - \bar{x})^2}{n - 1}}$? Wait no, the chart's standard deviation is $\sqrt{\frac{\sum (x - \bar{x})^2}{n}}$, which is population standard deviation. Wait the problem says "sample standard deviation" – maybe a mix. Wait let's check the chart:

The chart has:

- $\frac{\sum (x - \bar{x})^2}{n}$ for variance (maybe population variance)
- Standard Deviation: $\sqrt{\frac{\sum (x - \bar{x})^2}{n}}$ (population standard deviation) or $\sqrt{\frac{\sum (x - \bar{x})^2}{n - 1}}$ (sample). But the problem says "sample standard deviation" – so maybe n-1. Wait but the data: if it's five patients, but the data has 8 values, maybe a typo. Wait the original problem: "A doctor recorded the weight of five of his patients. The weight of each patient is: 29,20,21,19,35,28,33,31." Wait that's 8 values. Maybe the doctor has five patients but recorded 8 weights? No, that's confusing. Wait maybe the user made a typo, but let's proceed with the given data (8 values).

Wait the chart:

First, fill the x column with the data: 29,20,21,19,35,28,33,31 (8 rows)

$\bar{x}$ is 27 for each row.

Then $x - \bar{x}$: 2, -7, -6, -8, 8, 1, 6, 4

Then $(x - \bar{x})^2$: 4, 49, 36, 64, 64, 1, 36, 16

Sum of $(x - \bar{x})^2$: 270 (as calculated)

Then $\frac{\sum (x - \bar{x})^2}{n} = \frac{270}{8} = 33.75$ (variance)

Standard deviation: $\sqrt{\frac{270}{8}} = \sqrt{33.75} \approx 5.81$ (if population) or if sample, $\sqrt{\frac{270}{7}} \approx \sqrt{38.5714} \approx 6.21$. But the problem says "sample standard deviation" – so maybe n=8, sample standard deviation is $\sqrt{\frac{270}{7}} \approx 6.21$? Wait no, sample standard deviation uses n-1. Wait the chart's standard deviation formula is $\sqrt{\frac{\sum (x - \bar{x})^2}{n}}$, which is population. But the problem says "sample standard deviation" – maybe the chart is wrong, or the problem is mixed. Wait let's check the original problem again:

"Determine the variance of the patients' weights and the sample standard deviation."

So variance (population) is $\frac{\sum (x - \bar{x})^2}{n}$, sample standard deviation is $\sqrt{\frac{\sum (x - \bar{x})^2}{n - 1}}$.

Wait with n=8:

Population variance: 270 / 8 = 33.75

Sample variance: 270 / 7 ≈ 38.5714

Population standard deviation: √33.75 ≈ 5.81

Sample standard deviation: √(270/7) ≈ √38.5714 ≈ 6.21

But the chart has the standard deviation formula as $\sqrt{\frac{\sum (x - \bar{x})^2}{n}}$, so maybe the problem expects population standard deviation, but says sample. Maybe a mistake. Let's proceed with the chart's formula.

So:

- $\sum (x - \bar{x})^2 = 270$
- $\frac{\sum (x - \bar{x})^2}{n} = 270 / 8 = 33.75$ (variance)
- Standard deviation: √(270 / 8) = √33.75 ≈ 5.81 (rounded to nearest hundredth)

Wait but let's check the data again. Wait the sum of the weights: 29+20=49, +21=70, +19=89, +35=124, +28=152, +33=185, +31=216. 216 / 8 = 27. Correct. Then the squared differences:

29: 29-27=2, 4

20: 20-27=-7, 49

21: 21-27=-6, 36

19: 19-27=-8, 64

35: 35-27=8, 64

28: 28-27=1, 1

33: 33-27=6, 36

31: 31-27=4, 16

Sum: 4+49=53, +36=89, +64=153, +64=217, +1=218, +36=254, +16=270. Correct.

So filling the chart:

x | $\bar{x}$ | $x - \bar{x}$ | $(x - \bar{x})^2$

29 | 27 | 2 | 4

20 | 27 | -7 | 49

21 | 27 | -6 | 36

19 | 27 | -8 | 64

35 | 27 | 8 | 64

28 | 27 | 1 | 1

33 | 27 | 6 | 36

31 | 27 | 4 | 16

$\sum (x - \bar{x})^2$: 270

$\frac{\sum (x - \bar{x})^2}{n}$: 270 / 8 = 33.75

Standard Deviation: √(270 / 8) ≈ 5.81

Answer:

• $\sum (x - \bar{x})^2$: 270
• Variance ($\frac{\sum (x - \bar{x})^2}{n}$): 33.75
• Standard Deviation: $\approx 5.81$

(Note: If the problem intended 5 patients, there's a typo in the data. Assuming the given 8 values are correct, the above is the solution.)