QUESTION G:
To find the 91st percentile, we use the formula:
[tex]p=\frac{x-l}{h-l}[/tex]where
[tex]\begin{gathered} p=\text{ percentile} \\ l=\text{lowest value} \\ h=\text{ highest value} \\ x=91st\text{ percentile value} \end{gathered}[/tex]From the question, we have:
[tex]\begin{gathered} l=4 \\ h=36 \\ p=\frac{91}{100}=0.91 \end{gathered}[/tex]Therefore, we can calculate the value for x to be:
[tex]\begin{gathered} 0.91=\frac{x-4}{36-4} \\ 0.91=\frac{x-4}{32} \\ x-4=32\times0.91 \\ x-4=29.12 \\ x=29.12+4 \\ x=33.12 \end{gathered}[/tex]Therefore, the 91st percentile is 33.12
QUESTION H:
The lower quarter is the bottom 25% of the numbers.
The maximum of the bottom quarter is the 25th percentile.
Using the formula from the previous part, where x is now the 25th percentile value, we have the following parameters:
[tex]\begin{gathered} p=\frac{25}{100}=0.25 \\ l=4 \\ h=36 \end{gathered}[/tex]Therefore, the 25th percentile is calculated to be:
[tex]\begin{gathered} 0.25=\frac{x-4}{36-4} \\ 0.25=\frac{x-4}{32} \\ x-4=0.25\times32 \\ x-4=8 \\ x=8+4 \\ x=12 \end{gathered}[/tex]Therefore, the maximum value of the lower quarter is 12.
QUESTION F:
To calculate the probability that x > 5 given that x < 13, we can use the formula:
[tex]P=\frac{x_2-x_1}{h-l}[/tex]where, for x > 5
[tex]\begin{gathered} x_2=13 \\ x_1=5 \\ h=36 \\ l=4 \end{gathered}[/tex]Substituting into the formula, we have:
[tex]P(x>5)=\frac{13-5}{36-4}=0.25[/tex]and for x < 13
[tex]\begin{gathered} x_2=13 \\ x_1=4 \end{gathered}[/tex]Therefore:
[tex]P(x<13)=\frac{13-4}{32}=0.28125[/tex]Combining the probabilities, we have:
[tex]\begin{gathered} P(x>5|x<13)=\frac{P(x>5)}{P(x<13)} \\ P(x>5|x<13)=\frac{0.25}{0.28125} \\ P(x>5|x<13)=0.8889 \end{gathered}[/tex]