Question

1. now form a frequency distribution for the data. add more rows if needed. (3 points)

hours of sleep (x) frequency (f)
4 2
5 3
6 7
7 6
8 8
9 3
10 1
11 0
calculate descriptive statistics

1. calculate the following values. (you can use the ti - 83/84 or sagecalc.com) (9 points)

a. the mean $\bar{x}=$
b. the sample standard deviation $s=$
c. min =
d. max =
e. median =
f. $q_1=$
g. $q_3=$
h. iqr =
i. mode =

Explanation:

Step1: Calculate the mean

The formula for the mean $\bar{x}=\frac{\sum_{i = 1}^{n}x_if_i}{\sum_{i = 1}^{n}f_i}$. First, calculate $\sum_{i = 1}^{n}x_if_i$:
\[

$$\begin{align*} &4\times2 + 5\times3+6\times7 + 7\times6+8\times8+9\times3+10\times1+11\times0\\ =&8 + 15+42+42+64+27+10+0\\ =&208 \end{align*}$$

\]
$\sum_{i = 1}^{n}f_i=2 + 3+7+6+8+3+1+0=30$. So, $\bar{x}=\frac{208}{30}\approx6.93$.

Step2: Calculate the sample - standard deviation

The formula for the sample - standard deviation $s=\sqrt{\frac{\sum_{i = 1}^{n}f_i(x_i-\bar{x})^2}{n - 1}}$, where $n=\sum_{i = 1}^{n}f_i = 30$.
\[

$$\begin{align*} &\sum_{i = 1}^{n}f_i(x_i - \bar{x})^2\\ =&2\times(4 - 6.93)^2+3\times(5 - 6.93)^2+7\times(6 - 6.93)^2+6\times(7 - 6.93)^2+8\times(8 - 6.93)^2+3\times(9 - 6.93)^2+1\times(10 - 6.93)^2+0\times(11 - 6.93)^2\\ =&2\times(-2.93)^2+3\times(-1.93)^2+7\times(-0.93)^2+6\times(0.07)^2+8\times(1.07)^2+3\times(2.07)^2+1\times(3.07)^2+0\\ =&2\times8.5849+3\times3.7249+7\times0.8649+6\times0.0049+8\times1.1449+3\times4.2849+1\times9.4249\\ =&17.1698+11.1747+6.0543+0.0294+9.1592+12.8547+9.4249\\ =&65.867 \end{align*}$$

\]
$s=\sqrt{\frac{65.867}{29}}\approx1.50$.

Step3: Find the minimum value

From the data, the minimum value of hours of sleep is $Min = 4$.

Step4: Find the maximum value

From the data, the maximum value of hours of sleep is $Max = 10$.

Step5: Calculate the median

The total frequency $n = 30$. The median is the average of the 15th and 16th ordered values.
Cumulative frequencies:
For $x = 4$, $cf_1=2$; for $x = 5$, $cf_2=2 + 3=5$; for $x = 6$, $cf_3=5 + 7 = 12$; for $x = 7$, $cf_4=12+6 = 18$.
The 15th and 16th values fall in the $x = 7$ group, so the median $Median = 7$.

Step6: Calculate the first - quartile ($Q_1$)

$Q_1$ is the value at the $\frac{n}{4}=7.5$th ordered value.
Cumulative frequencies: $cf_1 = 2$, $cf_2=5$, $cf_3 = 12$. The 7.5th value falls in the $x = 6$ group, so $Q_1 = 6$.

Step7: Calculate the third - quartile ($Q_3$)

$Q_3$ is the value at the $\frac{3n}{4}=22.5$th ordered value.
Cumulative frequencies: $cf_1 = 2$, $cf_2=5$, $cf_3 = 12$, $cf_4=18$, $cf_5=18 + 8=26$. The 22.5th value falls in the $x = 8$ group, so $Q_3 = 8$.

Step8: Calculate the inter - quartile range

$IQR=Q_3 - Q_1=8 - 6 = 2$.

Step9: Calculate the mode

The mode is the value with the highest frequency. The highest frequency is 8, and the corresponding value of $x$ is 8. So, $Mode = 8$.

Answer:

a. $\bar{x}\approx6.93$
b. $s\approx1.50$
c. $Min = 4$
d. $Max = 10$
e. $Median = 7$
f. $Q_1 = 6$
g. $Q_3 = 8$
h. $IQR = 2$
i. $Mode = 8$