Question

6 from unit 1, lesson 13
a group of pennies made in 2018 are weighed. the mean is approximately 2.5 grams with a standard deviation of 0.02 grams.
interpret the mean and standard deviation in terms of the context.
7 from unit 1, lesson 12
these values represent the expected number of paintings a person will produce over the next 10 days.
0 0 0 1 1 1 2 2 3 5
a. what are the mean and standard deviation of the data?
mean:
standard deviation:
b. the artist is not pleased with these statistics. if the 5 is increased to a larger value, how does this affect the median, mean, and standard deviation?

Explanation:

Step1: Calculate the mean of the painting - data

The formula for the mean $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$, where $n = 10$ and $x_{i}$ are the data - points.
$\sum_{i=1}^{10}x_{i}=0 + 0+0 + 1+1+1+2+2+3+5=15$
$\bar{x}=\frac{15}{10}=1.5$

Step2: Calculate the variance of the painting - data

The formula for the variance $s^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}$.
$(0 - 1.5)^{2}=2.25$, and there are 3 zeros, so the contribution is $3\times2.25 = 6.75$.
$(1 - 1.5)^{2}=0.25$, and there are 3 ones, so the contribution is $3\times0.25 = 0.75$.
$(2 - 1.5)^{2}=0.25$, and there are 2 twos, so the contribution is $2\times0.25 = 0.5$.
$(3 - 1.5)^{2}=2.25$, and there is 1 three, so the contribution is $2.25$.
$(5 - 1.5)^{2}=12.25$, and there is 1 five, so the contribution is $12.25$.
$\sum_{i = 1}^{10}(x_{i}-1.5)^{2}=6.75+0.75 + 0.5+2.25+12.25=22.5$
$s^{2}=\frac{22.5}{9}=2.5$

Step3: Calculate the standard deviation of the painting - data

The standard deviation $s=\sqrt{s^{2}}$, so $s=\sqrt{2.5}\approx1.58$

Step4: Analyze the effect of increasing 5 to a larger value on median, mean, and standard - deviation

The data set in ascending order is $0,0,0,1,1,1,2,2,3,5$. The median is the average of the 5th and 6th ordered values, so the median is $\frac{1 + 1}{2}=1$.

• Median: Since the median is based on the middle - value(s) of the ordered data set, increasing the largest value (5) will not affect the median as long as the number of data points remains 10 and the middle - two values do not change. So the median remains the same.
• Mean: The mean is $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$. If we increase the value of 5 to a larger value, $\sum_{i = 1}^{n}x_{i}$ will increase, and since $n$ is fixed ($n = 10$), the mean will increase.
• Standard deviation: The standard deviation measures the spread of the data. Increasing the largest value will increase the spread of the data, so the standard deviation will increase.

For question 6:
The mean of 2.5 grams for the pennies means that, on average, a penny made in 2018 in this group weighs approximately 2.5 grams. The standard deviation of 0.02 grams indicates that the weights of the pennies in the group typically vary from the mean weight of 2.5 grams by about 0.02 grams.

For question 7:

• a. Mean: 1.5
• a. Standard deviation: $\approx1.58$
• b. Median: Remains the same.
• b. Mean: Increases.
• b. Standard deviation: Increases.

Answer:

Step1: Calculate the mean of the painting - data

The formula for the mean $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$, where $n = 10$ and $x_{i}$ are the data - points.
$\sum_{i=1}^{10}x_{i}=0 + 0+0 + 1+1+1+2+2+3+5=15$
$\bar{x}=\frac{15}{10}=1.5$

Step2: Calculate the variance of the painting - data

The formula for the variance $s^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}$.
$(0 - 1.5)^{2}=2.25$, and there are 3 zeros, so the contribution is $3\times2.25 = 6.75$.
$(1 - 1.5)^{2}=0.25$, and there are 3 ones, so the contribution is $3\times0.25 = 0.75$.
$(2 - 1.5)^{2}=0.25$, and there are 2 twos, so the contribution is $2\times0.25 = 0.5$.
$(3 - 1.5)^{2}=2.25$, and there is 1 three, so the contribution is $2.25$.
$(5 - 1.5)^{2}=12.25$, and there is 1 five, so the contribution is $12.25$.
$\sum_{i = 1}^{10}(x_{i}-1.5)^{2}=6.75+0.75 + 0.5+2.25+12.25=22.5$
$s^{2}=\frac{22.5}{9}=2.5$

Step3: Calculate the standard deviation of the painting - data

The standard deviation $s=\sqrt{s^{2}}$, so $s=\sqrt{2.5}\approx1.58$

Step4: Analyze the effect of increasing 5 to a larger value on median, mean, and standard - deviation

The data set in ascending order is $0,0,0,1,1,1,2,2,3,5$. The median is the average of the 5th and 6th ordered values, so the median is $\frac{1 + 1}{2}=1$.

• Median: Since the median is based on the middle - value(s) of the ordered data set, increasing the largest value (5) will not affect the median as long as the number of data points remains 10 and the middle - two values do not change. So the median remains the same.
• Mean: The mean is $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$. If we increase the value of 5 to a larger value, $\sum_{i = 1}^{n}x_{i}$ will increase, and since $n$ is fixed ($n = 10$), the mean will increase.
• Standard deviation: The standard deviation measures the spread of the data. Increasing the largest value will increase the spread of the data, so the standard deviation will increase.

For question 6:
The mean of 2.5 grams for the pennies means that, on average, a penny made in 2018 in this group weighs approximately 2.5 grams. The standard deviation of 0.02 grams indicates that the weights of the pennies in the group typically vary from the mean weight of 2.5 grams by about 0.02 grams.

For question 7:

• a. Mean: 1.5
• a. Standard deviation: $\approx1.58$
• b. Median: Remains the same.
• b. Mean: Increases.
• b. Standard deviation: Increases.