1. what is the standard deviation of a distribution of numbers?
a.) a measurement of how far each observation is from the closest other observation, on average
b.) the distance between the max and the min
c.) the average distance each observation in a distribution is away from the mean of that distribution
d.) a measurement of where the center of a distribution of numbers falls
2. if the standard deviation of a distribution is 0, what must be true about that distribution?
a.) all numbers have to be 0
b.) all numbers have to be the same
c.) the distribution must be approximately normal
d.) there is no difference between the q3 and q1
3. if the standard deviation of a distribution is 1, what must be true about that distribution?
a.) all numbers have to be 1
b.) all numbers have to be the same
c.) all of the numbers average about 1 away from the mean
d.) the distribution is skewed
4. what is the standard deviation of the following list of numbers? (sigma)
4 14 6 13 5 12 15 9 25 22.5
a.) about 5.4
b.) about 6.7
c.) about 8.6
d.) about 9.2
5. which of the following distributions would have the largest standard deviation?
a.) the age in years of students in the room
b.) the price of homes sold in kyle this month
c.) the number of pets owned by students at hhs
d.) the grades on this quiz
6. the distribution of 1^st nine weeks grades in statistics is normally distributed with a mean of 60 and a standard deviation of 7.
what proportion of grades are between 60 and 70?
a.).3557
b.).4234
c.).5767
d.).6221

Question

Accepted Answer

# 1.
## Brief Explanations:
The standard deviation measures the average distance each observation is from the mean. Option a is incorrect as it mentions distance from the closest observation. Option b is the range. Option d is about central tendency (not standard deviation).
## Answer:
c. The average distance each observation in a distribution is away from the mean of that distribution

# 2.
## Brief Explanations:
The formula for standard deviation $\sigma=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\mu)^{2}}{n}}$. If $\sigma = 0$, then $(x_{i}-\mu)^{2}=0$ for all $i$, which means $x_{i}=\mu$ for all $i$ (all numbers are the same). Option a is wrong (e.g., $2,2,2$ has $\sigma=0$). Option c: normal distribution has $\sigma>0$ usually. Option d: Q3 - Q1 is the inter - quartile range.
## Answer:
b. All numbers have to be the same

# 3.
## Brief Explanations:
Standard deviation of 1 means, on average, each number is about 1 unit away from the mean. Option a: e.g., $0,2$ has $\sigma = 1$. Option b: if all numbers are the same $\sigma=0$. Option d: skewness is about asymmetry, not directly related to $\sigma = 1$.
## Answer:
c. All of the numbers average about 1 away from the mean

# 4.
## Step - by - Step Format:
### Step1: Calculate the mean $\mu$
$$
\begin{align*}
\mu&=\frac{4 + 14+6+13+5+12+15+9+25+22.5}{10}\
&=\frac{125.5}{10}=12.55
\end{align*}
$$
### Step2: Calculate $(x_{i}-\mu)^{2}$
$$
\begin{align*}
(4 - 12.55)^{2}&=(- 8.55)^{2}=73.1025\
(14 - 12.55)^{2}&=(1.45)^{2}=2.1025\
(6 - 12.55)^{2}&=(-6.55)^{2}=42.9025\
(13 - 12.55)^{2}&=(0.45)^{2}=0.2025\
(5 - 12.55)^{2}&=(-7.55)^{2}=57.0025\
(12 - 12.55)^{2}&=(-0.55)^{2}=0.3025\
(15 - 12.55)^{2}&=(2.45)^{2}=6.0025\
(9 - 12.55)^{2}&=(-3.55)^{2}=12.6025\
(25 - 12.55)^{2}&=(12.45)^{2}=155.0025\
(22.5 - 12.55)^{2}&=(9.95)^{2}=99.0025
\end{align*}
$$
### Step3: Calculate the variance $\sigma^{2}$
$$
\begin{align*}
\sigma^{2}&=\frac{73.1025 + 2.1025+42.9025+0.2025+57.0025+0.3025+6.0025+12.6025+155.0025+99.0025}{10}\
&=\frac{448.225}{10}=44.8225
\end{align*}
$$
### Step4: Calculate the standard deviation $\sigma$
$$
\sigma=\sqrt{44.8225}\approx6.7
$$
## Answer:
b. about 6.7

# 5.
## Brief Explanations:
Home prices ($b$) are likely to have a larger spread (more variability) compared to student ages (which are relatively close in a class), number of pets (limited range), and quiz grades (usually within a certain range like $0 - 100$).
## Answer:
b. The price of homes sold in Kyle this month

# 6.
## Step - by - Step Format:
### Step1: Calculate the z - score
The z - score formula is $z=\frac{x-\mu}{\sigma}$. For $x = 70$, $\mu = 60$, $\sigma=7$
$$
z=\frac{70 - 60}{7}=\frac{10}{7}\approx1.43
$$
### Step2: Use the standard normal table
The proportion of values to the left of $z = 1.43$ is $P(Z<1.43)=0.9236$, and the proportion of values to the left of $z = 0$ (since $\mu = 60$, $x = 60$ gives $z = 0$) is $P(Z < 0)=0.5$
### Step3: Calculate the proportion between $x = 60$ and $x = 70$
$$
P(60<X<70)=P(0<Z<1.43)=P(Z < 1.43)-P(Z < 0)=0.9236-0.5 = 0.4236\approx0.4234
$$
## Answer:
b..4234

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