Question

score on last try: 0 of 3 pts. see details for more. at least one scored part is incorrect. jump to first changable incorrect part. get a similar question you can retry this question below this data is from a sample. calculate the mean and standard deviation. x 39.8 45.8 30.7 11.9 48.4 26.5 19.4 26.7 please show the following answers to 1 or 2 decimal places. sample mean = 49.5 × sample standard deviation = 54 × ooops - now you discover that the data was actually from a population! so now you must give the population standard deviation. (this one must be to 2 decimal places.) population standard deviation = 15.18 × question help: message instructor post to forum

Explanation:

Step1: Calculate the sample mean

The formula for the mean (for both sample and population) 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.
First, sum the data points: $39.8 + 45.8 + 30.7 + 11.9 + 48.4 + 26.5 + 19.4 + 26.7$.
Let's calculate the sum:
$39.8+45.8 = 85.6$; $85.6+30.7 = 116.3$; $116.3+11.9 = 128.2$; $128.2+48.4 = 176.6$; $176.6+26.5 = 203.1$; $203.1+19.4 = 222.5$; $222.5+26.7 = 249.2$.
There are $n = 8$ data points. So the mean $\bar{x}=\frac{249.2}{8}=31.15$.

Step2: Calculate the sample standard deviation

The formula for sample standard deviation $s$ is $s=\sqrt{\frac{\sum_{i = 1}^{n}(x_i - \bar{x})^2}{n - 1}}$.
First, calculate each $(x_i - \bar{x})^2$:

• For $x_1 = 39.8$: $(39.8 - 31.15)^2=(8.65)^2 = 74.8225$
• For $x_2 = 45.8$: $(45.8 - 31.15)^2=(14.65)^2 = 214.6225$
• For $x_3 = 30.7$: $(30.7 - 31.15)^2=(- 0.45)^2 = 0.2025$
• For $x_4 = 11.9$: $(11.9 - 31.15)^2=(-19.35)^2 = 374.4225$
• For $x_5 = 48.4$: $(48.4 - 31.15)^2=(17.25)^2 = 297.5625$
• For $x_6 = 26.5$: $(26.5 - 31.15)^2=(-4.65)^2 = 21.6225$
• For $x_7 = 19.4$: $(19.4 - 31.15)^2=(-11.75)^2 = 138.0625$
• For $x_8 = 26.7$: $(26.7 - 31.15)^2=(-4.45)^2 = 19.8025$

Now sum these squared differences:
$74.8225+214.6225 = 289.445$; $289.445+0.2025 = 289.6475$; $289.6475+374.4225 = 664.07$; $664.07+297.5625 = 961.6325$; $961.6325+21.6225 = 983.255$; $983.255+138.0625 = 1121.3175$; $1121.3175+19.8025 = 1141.12$.

Now, $n - 1=7$, so $\frac{\sum(x_i - \bar{x})^2}{n - 1}=\frac{1141.12}{7}\approx163.0171$.
Then $s=\sqrt{163.0171}\approx12.77$.

Step3: Calculate the population standard deviation

The formula for population standard deviation $\sigma$ is $\sigma=\sqrt{\frac{\sum_{i = 1}^{n}(x_i - \bar{x})^2}{n}}$.
We already calculated $\sum(x_i - \bar{x})^2 = 1141.12$ and $n = 8$. So $\frac{\sum(x_i - \bar{x})^2}{n}=\frac{1141.12}{8}=142.64$.
Then $\sigma=\sqrt{142.64}\approx11.94$.

Answer:

Sample Mean = $31.15$
Sample Standard Deviation = $12.77$ (approx to two decimal places)
Population Standard Deviation = $11.94$