4. now form a frequency distribution for the data. add more rows if needed. (3 points)\nhours of sleep (x)…

4. now form a frequency distribution for the data. add more rows if needed. (3 points)\nhours of sleep (x) frequency (f)\n4 2\n5 3\n6 7\n7 6\n8 8\n9 3\n10 1\n11 0\ncalculate descriptive statistics\n7. calculate the following values. (you can use the ti - 83/84 or sagecalc.com) (9 points)\na. the mean $\bar{x}=$ \nb. the sample standard deviation $s=$ \nc. min = \nd. max = \ne. median = \nf. $q_1=$ \ng. $q_3=$ \nh. iqr = \ni. mode =

4. now form a frequency distribution for the data. add more rows if needed. (3 points)\nhours of sleep (x) frequency (f)\n4 2\n5 3\n6 7\n7 6\n8 8\n9 3\n10 1\n11 0\ncalculate descriptive statistics\n7. calculate the following values. (you can use the ti - 83/84 or sagecalc.com) (9 points)\na. the mean $\bar{x}=$ \nb. the sample standard deviation $s=$ \nc. min = \nd. max = \ne. median = \nf. $q_1=$ \ng. $q_3=$ \nh. iqr = \ni. mode =

Answer

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. $Mode = 8$