A factory manufactures tubes. Part A Questions 1. and 2. are independent. We are interested in two types of tubes, called type 1 tubes and type 2 tubes.
A type 1 tube is accepted at inspection if its thickness is between 1.35 millimetres and 1.65 millimetres. a. Let $X$ denote the random variable that, for each type 1 tube randomly selected from the day's production, gives its thickness expressed in millimetres. We assume that the random variable $X$ follows a normal distribution with mean 1.5 and standard deviation 0.07. A type 1 tube is randomly selected from the day's production. Calculate the probability that the tube is accepted at inspection. b. The company wishes to improve the quality of type 1 tube production. To do this, the settings of the machines producing these tubes are modified. Let $X _ { 1 }$ denote the random variable that, for each type 1 tube selected from the production of the modified machine, gives its thickness. We assume that the random variable $X _ { 1 }$ follows a normal distribution with mean 1.5 and standard deviation $\sigma _ { 1 }$. A type 1 tube is randomly selected from the production of the modified machine. Determine an approximate value to $10 ^ { - 3 }$ of $\sigma _ { 1 }$ so that the probability that this tube is accepted at inspection equals 0.98. (You may use the random variable $Z$ defined by $Z = \frac { X _ { 1 } - 1.5 } { \sigma _ { 1 } }$ which follows the standard normal distribution.)
A machine produces type 2 tubes. A type 2 tube is said to be ``compliant for length'' when its length, in millimetres, belongs to the interval [298; 302]. The specifications establish that, in the production of type 2 tubes, a proportion of $2 \%$ of tubes that are not ``compliant for length'' is acceptable. It is desired to decide whether the production machine should be serviced. To do this, a random sample of 250 tubes is taken from the production of type 2 tubes, in which 10 tubes are found to be not ``compliant for length''. a. Give an asymptotic confidence interval at $95 \%$ for the frequency of tubes not ``compliant for length'' in a sample of 250 tubes. b. Should the machine be serviced? Justify your answer.
Part B Adjustment errors in the production line can affect the thickness or length of type 2 tubes. A study conducted on the production revealed that:
$96 \%$ of type 2 tubes have compliant thickness;
among type 2 tubes that have compliant thickness, $95 \%$ have compliant length;
$3.6 \%$ of type 2 tubes have non-compliant thickness and compliant length.
A type 2 tube is randomly selected from the production and we consider the events: --- $E$: ``the thickness of the tube is compliant''; --- $L$: ``the length of the tube is compliant''. We model the random experiment with a probability tree.
Copy and complete this tree entirely.
Show that the probability of event $L$ equals 0.948.
A factory manufactures tubes.
\textbf{Part A}
Questions 1. and 2. are independent.\\
We are interested in two types of tubes, called type 1 tubes and type 2 tubes.
\begin{enumerate}
\item A type 1 tube is accepted at inspection if its thickness is between 1.35 millimetres and 1.65 millimetres.\\
a. Let $X$ denote the random variable that, for each type 1 tube randomly selected from the day's production, gives its thickness expressed in millimetres. We assume that the random variable $X$ follows a normal distribution with mean 1.5 and standard deviation 0.07.
A type 1 tube is randomly selected from the day's production. Calculate the probability that the tube is accepted at inspection.\\
b. The company wishes to improve the quality of type 1 tube production. To do this, the settings of the machines producing these tubes are modified. Let $X _ { 1 }$ denote the random variable that, for each type 1 tube selected from the production of the modified machine, gives its thickness. We assume that the random variable $X _ { 1 }$ follows a normal distribution with mean 1.5 and standard deviation $\sigma _ { 1 }$.
A type 1 tube is randomly selected from the production of the modified machine. Determine an approximate value to $10 ^ { - 3 }$ of $\sigma _ { 1 }$ so that the probability that this tube is accepted at inspection equals 0.98. (You may use the random variable $Z$ defined by $Z = \frac { X _ { 1 } - 1.5 } { \sigma _ { 1 } }$ which follows the standard normal distribution.)
\item A machine produces type 2 tubes. A type 2 tube is said to be ``compliant for length'' when its length, in millimetres, belongs to the interval [298; 302]. The specifications establish that, in the production of type 2 tubes, a proportion of $2 \%$ of tubes that are not ``compliant for length'' is acceptable.
It is desired to decide whether the production machine should be serviced. To do this, a random sample of 250 tubes is taken from the production of type 2 tubes, in which 10 tubes are found to be not ``compliant for length''.\\
a. Give an asymptotic confidence interval at $95 \%$ for the frequency of tubes not ``compliant for length'' in a sample of 250 tubes.\\
b. Should the machine be serviced? Justify your answer.
\end{enumerate}
\textbf{Part B}
Adjustment errors in the production line can affect the thickness or length of type 2 tubes.\\
A study conducted on the production revealed that:
\begin{itemize}
\item $96 \%$ of type 2 tubes have compliant thickness;
\item among type 2 tubes that have compliant thickness, $95 \%$ have compliant length;
\item $3.6 \%$ of type 2 tubes have non-compliant thickness and compliant length.
\end{itemize}
A type 2 tube is randomly selected from the production and we consider the events:\\
--- $E$: ``the thickness of the tube is compliant'';\\
--- $L$: ``the length of the tube is compliant''.\\
We model the random experiment with a probability tree.
\begin{enumerate}
\item Copy and complete this tree entirely.
\item Show that the probability of event $L$ equals 0.948.
\end{enumerate}