The calcium content in one bottle of beverage produced by a certain company follows a normal distribution with population mean $m$ and population standard deviation $\sigma$. When 16 bottles of beverage produced by this company were randomly sampled and the calcium content was measured, the sample mean was 12.34. When the 95\% confidence interval for the population mean $m$ of the calcium content in one bottle of beverage produced by this company is $11.36 \leq m \leq a$, what is the value of $a + \sigma$? (Note: When $Z$ follows the standard normal distribution, $\mathrm { P } ( 0 \leq Z \leq 1.96 ) = 0.4750$, and the unit of calcium content is mg.) [3 points]
(1) 14.32
(2) 14.82
(3) 15.32
(4) 15.82
(5) 16.32

The daily leisure activity time of residents in a certain region follows a normal distribution with mean $m$ minutes and standard deviation $\sigma$ minutes. When 16 residents are randomly sampled and the sample mean of daily leisure activity time is 75 minutes, the 95\% confidence interval for the population mean $m$ is $a \leq m \leq b$. When 16 residents are randomly sampled again and the sample mean of daily leisure activity time is 77 minutes, the 99\% confidence interval for the population mean $m$ is $c \leq m \leq d$. Find the value of $\sigma$ that satisfies $d - b = 3.86$. (Here, when $Z$ is a random variable following the standard normal distribution, $\mathrm { P } ( | Z | \leq 1.96 ) = 0.95 , \mathrm { P } ( | Z | \leq 2.58 ) = 0.99$.) [4 points]

A certain automobile company produces electric vehicles whose driving range on a single charge follows a normal distribution with mean $m$ and standard deviation $\sigma$.
When a sample of 100 electric vehicles produced by this company is randomly selected and the sample mean of the driving range is $\overline { x _ { 1 } }$, the 95\% confidence interval for the population mean $m$ is $a \leq m \leq b$.
When a sample of 400 electric vehicles produced by this company is randomly selected and the sample mean of the driving range is $\overline { x _ { 2 } }$, the 99\% confidence interval for the population mean $m$ is $c \leq m \leq d$.
If $\overline { x _ { 1 } } - \overline { x _ { 2 } } = 1.34$ and $a = c$, what is the value of $b - a$? (Here, the unit of driving range is km, and when $Z$ is a random variable following the standard normal distribution, $\mathrm { P } ( | Z | \leq 1.96 ) = 0.95 , \mathrm { P } ( | Z | \leq 2.58 ) = 0.99$.) [3 points]
(1) 5.88
(2) 7.84
(3) 9.80
(4) 11.76
(5) 13.72