17. (12 points) To understand the retention levels of two types of ions in mice, the following experiment was conducted: 200 mice were randomly divided into groups $A$ and $B$, with 100 mice in each group. Group $A$ mice were given a solution of ion type 1, and group $B$ mice were given a solution of ion type 2. Each mouse was given the same volume of solution with the same molar concentration. After a period of time, the percentage of ions retained in the mice's bodies was measured using a scientific method. Based on the experimental data, the following histograms were obtained: [Figure][Figure] Let $C$ be the event: ``the retention percentage of ion type 2 in the body is at least 5.5.'' Based on the histogram, the estimated value of $P ( C )$ is 0.70. (1) Find the values of $a$ and $b$ in the histogram for ion type 2's retention percentage. (2) Estimate the mean retention percentages for ions of type 1 and type 2 respectively (use the midpoint of each interval as the representative value for data in that interval).
Solution:
17. (12 points)\\
To understand the retention levels of two types of ions in mice, the following experiment was conducted: 200 mice were randomly divided into groups $A$ and $B$, with 100 mice in each group. Group $A$ mice were given a solution of ion type 1, and group $B$ mice were given a solution of ion type 2. Each mouse was given the same volume of solution with the same molar concentration. After a period of time, the percentage of ions retained in the mice's bodies was measured using a scientific method. Based on the experimental data, the following histograms were obtained:\\
\includegraphics[max width=\textwidth, alt={}, center]{a4af1979-49a3-49ee-a522-f6f2c82dc334-04_449_721_760_310}\\
\includegraphics[max width=\textwidth, alt={}, center]{a4af1979-49a3-49ee-a522-f6f2c82dc334-04_451_725_753_1041}
Let $C$ be the event: ``the retention percentage of ion type 2 in the body is at least 5.5.'' Based on the histogram, the estimated value of $P ( C )$ is 0.70.\\
(1) Find the values of $a$ and $b$ in the histogram for ion type 2's retention percentage.\\
(2) Estimate the mean retention percentages for ions of type 1 and type 2 respectively (use the midpoint of each interval as the representative value for data in that interval).\\