2. the variance of a distribution is 50. what is the standard deviation, rounded to the nearest thousandth? 3. the following is a list of relay for life donations given by several community businesses in the maple glen high school vicinity. $10, $50, $100, $100, $100, $120, $120, $125, $150, $150, $250. round your answers to the nearest hundredth. a. what is the mean donation amount? b. what is the range? c. what is the variance? d. what is the standard deviation?

Question

Accepted Answer

### Problem 2
# Explanation:
## Step1: Recall the formula for standard deviation from variance.
The standard deviation ($\sigma$) is the square root of the variance ($\sigma^2$). So, if variance $\sigma^2 = 50$, then standard deviation $\sigma=\sqrt{50}$.
## Step2: Calculate the square root of 50 and round to the nearest thousandth.
$\sqrt{50}\approx7.0710678\approx7.071$
# Answer:
7.071

### Problem 3
#### Part a: Mean Donation Amount
# Explanation:
## Step1: Recall the formula for the mean.
The mean ($\bar{x}$) of a set of numbers is the sum of the numbers divided by the count of numbers. The formula is $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$, where $x_{i}$ are the data points and $n$ is the number of data points.
## Step2: Calculate the sum of the donation amounts.
The donations are $10, 50, 100, 100, 100, 120, 120, 125, 150, 150, 250$.
Sum $S=10 + 50+100+100+100+120+120+125+150+150+250$
$S = 10+50 = 60$; $60+100 = 160$; $160+100 = 260$; $260+100 = 360$; $360+120 = 480$; $480+120 = 600$; $600+125 = 725$; $725+150 = 875$; $875+150 = 1025$; $1025+250 = 1275$
## Step3: Determine the number of data points ($n$).
There are 11 data points.
## Step4: Calculate the mean.
$\bar{x}=\frac{1275}{11}\approx115.91$ (rounded to the nearest hundredth)
# Answer:
115.91

#### Part b: Range
# Explanation:
## Step1: Recall the formula for the range.
The range is the difference between the maximum value and the minimum value in the data set.
## Step2: Identify the maximum and minimum values.
The minimum value is $10$ and the maximum value is $250$.
## Step3: Calculate the range.
Range $=250 - 10=240$
# Answer:
240

#### Part c: Variance
# Explanation:
## Step1: Recall the formula for population variance (since this is a complete set of donations from the businesses in the vicinity, we use population variance).
The population variance ($\sigma^2$) is given by $\sigma^2=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^2}{n}$, where $\bar{x}$ is the mean, $x_{i}$ are the data points, and $n$ is the number of data points.
## Step2: Calculate $(x_{i}-\bar{x})^2$ for each data point.
We know $\bar{x}\approx115.9091$ (more precise value for calculation) and $n = 11$.
- For $x = 10$: $(10 - 115.9091)^2=(- 105.9091)^2\approx11216.74$
- For $x = 50$: $(50 - 115.9091)^2=(-65.9091)^2\approx4344.01$
- For $x = 100$ (three times): $(100 - 115.9091)^2=(-15.9091)^2\approx253.10$, so for three points: $3	imes253.10 = 759.30$
- For $x = 120$ (two times): $(120 - 115.9091)^2=(4.0909)^2\approx16.74$, so for two points: $2	imes16.74 = 33.48$
- For $x = 125$: $(125 - 115.9091)^2=(9.0909)^2\approx82.68$
- For $x = 150$ (two times): $(150 - 115.9091)^2=(34.0909)^2\approx1162.18$, so for two points: $2	imes1162.18 = 2324.36$
- For $x = 250$: $(250 - 115.9091)^2=(134.0909)^2\approx18000.56$
## Step3: Sum up all the squared differences.
Sum $=11216.74+4344.01 + 759.30+33.48+82.68+2324.36+18000.56$
$11216.74+4344.01=15560.75$; $15560.75 + 759.30=16320.05$; $16320.05+33.48 = 16353.53$; $16353.53+82.68 = 16436.21$; $16436.21+2324.36 = 18760.57$; $18760.57+18000.56 = 36761.13$
## Step4: Calculate the variance.
$\sigma^2=\frac{36761.13}{11}\approx3341.92$ (rounded to the nearest hundredth)
# Answer:
3341.92

#### Part d: Standard Deviation
# Explanation:
## Step1: Recall the formula for standard deviation from variance.
The standard deviation ($\sigma$) is the square root of the variance ($\sigma^2$). So, $\sigma=\sqrt{\sigma^2}$
## Step2: Calculate the square root of the variance.
We know $\sigma^2\approx3341.92$, so $\sigma=\sqrt{3341.92}\approx57.81$ (rounded to the nearest hundredth)
# Answer:
57.81

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Problem 2

Step1: Recall the formula for standard deviation from variance.

Step2: Calculate the square root of 50 and round to the nearest thousandth.

Step1: Recall the formula for the mean.

Step2: Calculate the sum of the donation amounts.

Step3: Determine the number of data points ($n$).

Step4: Calculate the mean.

Step1: Recall the formula for the range.

Step2: Identify the maximum and minimum values.

Step3: Calculate the range.

Answer:

Problem 3

Part a: Mean Donation Amount