Question

data collection:
data table 1: number of drops of distilled water contained on the surface of a penny
trial # drops distilled water # drops distilled water + soap
1 31 8
2 15 14
3 23 17
4 29 13
5 26 14
average 25 13
data analysis: do the appropriate calculations in data table 2 below.
data table 2: statistical analysis of the number of drops of distilled water contained on the surface of a penny
calculation # drops distilled water # drops distilled water + soap
mean 25 13
standard deviation 6.26 3.17
+/- 1 std dev
+/- 2 std dev
standard error 6 3
+/- 2 sem
s√(σ(x - x̄)² / n - 1) n = 5
31 - 25=(6)² = 36
15 - 25=(-10)² = 100
23 - 25=(-2)² = 4
29 - 25=(4)² = 16
26 - 25=(1)² = 1
ς = 167
124÷5
x̄ = 24.8
or
7:25
↑
mean
so
sex = 6.26 / √5
sex = 6.26 / 2.23
sex̄ = 2.80
or
2.8×2
5.6
or
↑
se
8 - 13=(-5)² = 25
14 - 13=(1)² = 1
17 - 13=(4)² = 16
13 - 13=(0)² = 0
14 - 13=(1)² = 1
ς = 43
66÷5
x̄ = 13.2
or
13
↑
mean
n = 5
sex = 3.17 / √5
sex = 3.17 / 2.23
sex̄ = 1.42
or
1.4×2
3
so

Explanation:

Step1: Recall the formula for standard - deviation

The formula for the sample standard deviation is $s=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}}$, where $x_{i}$ are the data points, $\bar{x}$ is the mean, and $n$ is the number of data points.

Step2: Calculate $\pm1$ standard deviation for distilled water

For distilled water, mean $\bar{x}=25$ and standard deviation $s = 6.26$.
The lower limit of $\pm1$ standard deviation is $25-6.26=18.74$ and the upper limit is $25 + 6.26=31.26$.

Step3: Calculate $\pm2$ standard deviation for distilled water

The lower limit of $\pm2$ standard deviation is $25-2\times6.26=25 - 12.52 = 12.48$ and the upper limit is $25+2\times6.26=25 + 12.52=37.52$.

Step4: Calculate $\pm1$ standard deviation for distilled water + soap

For distilled water + soap, mean $\bar{x}=13$ and standard deviation $s = 3.17$.
The lower limit of $\pm1$ standard deviation is $13-3.17 = 9.83$ and the upper limit is $13+3.17=16.17$.

Step5: Calculate $\pm2$ standard deviation for distilled water + soap

The lower limit of $\pm2$ standard deviation is $13-2\times3.17=13 - 6.34 = 6.66$ and the upper limit is $13+2\times3.17=13 + 6.34 = 19.34$.

Step6: Recall the formula for standard error

The formula for the standard error of the mean (SEM) is $SEM=\frac{s}{\sqrt{n}}$, where $s$ is the standard deviation and $n$ is the number of data points. For $n = 5$, for distilled water $SEM=\frac{6.26}{\sqrt{5}}\approx2.80$ and for distilled water + soap $SEM=\frac{3.17}{\sqrt{5}}\approx1.42$.

Step7: Calculate $\pm2$ SEM for distilled water

The lower limit of $\pm2$ SEM is $25-2\times2.80=25 - 5.6=19.4$ and the upper limit is $25+2\times2.80=25 + 5.6 = 30.6$.

Step8: Calculate $\pm2$ SEM for distilled water + soap

The lower limit of $\pm2$ SEM is $13-2\times1.42=13 - 2.84 = 10.16$ and the upper limit is $13+2\times1.42=13 + 2.84=15.84$.

We fill in Data Table 2 as follows:

Calculation	# Drops Distilled Water	# Drops Distilled Water + Soap
$\pm2$ std dev	$12.48 - 37.52$	$6.66 - 19.34$
$\pm2$ SEM	$19.4 - 30.6$	$10.16 - 15.84$

Answer:

Calculation	# Drops Distilled Water	# Drops Distilled Water + Soap
$\pm2$ std dev	$12.48 - 37.52$	$6.66 - 19.34$
$\pm2$ SEM	$19.4 - 30.6$	$10.16 - 15.84$