In an automobile factory, certain metal parts are covered with a thin layer of nickel that protects them against corrosion and wear. The process used is electroplating with nickel.
It is assumed that the random variable $X$, which associates to each treated part the thickness of nickel deposited, follows a normal distribution with mean $\mu _ { 1 } = 25$ micrometers ( $\mu \mathrm { m }$ ) and standard deviation $\sigma _ { 1 }$.
A part is compliant if the thickness of nickel deposited is between $22.8 \mu \mathrm {~m}$ and $27.2 \mu \mathrm {~m}$.
The probability density function of $X$ is represented below. It was determined that $P ( X > 27.2 ) = 0.023$.

1. a. Determine the probability that a part is compliant. b. Justify that 1.1 is an approximate value of $\sigma _ { 1 }$ to within $10 ^ { - 1 }$. c. Given that a part is compliant, calculate the probability that the thickness of nickel deposited on it is less than $24 \mu \mathrm {~m}$. Round to $10 ^ { - 3 }$.
2. A team of engineers proposes another nickel plating process, obtained by chemical reaction without any current source. The team claims that this new process theoretically allows obtaining $98 \%$ of compliant parts. The random variable $Y$ which, for each part treated with this new process, associates the thickness of nickel deposited follows a normal distribution with mean $\mu _ { 2 } = 25 \mu \mathrm {~m}$ and standard deviation $\sigma _ { 2 }$. a. Assuming the above claim, compare $\sigma _ { 1 }$ and $\sigma _ { 2 }$. b. A quality control evaluates the new process; it reveals that out of 500 parts tested, 15 are not compliant. At the $95 \%$ confidence level, can we reject the team's claim?