SPS SPS SM Statistics 2024 January — Question 2 14 marks

Exam BoardSPS
ModuleSPS SM Statistics (SPS SM Statistics)
Year2024
SessionJanuary
Marks14
TopicNormal Distribution
TypeValidity of normal model
DifficultyModerate -0.8 This is a routine A-level statistics question involving histogram interpretation, normal distribution calculations, and model comparison. Part (a) requires basic frequency density reading, parts (b)-(d) involve standard normal probability calculations with given parameters, and part (e) requires comparing models—all standard textbook exercises with no novel problem-solving or conceptual challenges.
Spec2.02b Histogram: area represents frequency2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation
The heights, in centimetres, of a random sample of 150 plants of a certain variety were measured. The results are summarised in the histogram. \includegraphics{figure_2} One of the 150 plants is chosen at random, and its height, \(X\) cm, is noted.
  1. Show that P\((20 < X < 30) = 0.147\), correct to 3 significant figures. [2]
Sam suggests that the distribution of \(X\) can be well modelled by the distribution N\((40, 100)\).
    1. Give a brief justification for the use of the normal distribution in this context. [1]
    2. Give a brief justification for the choice of the parameter values 40 and 100. [2]
  2. Use Sam's model to find P\((20 < X < 30)\). [1]
Nina suggests a different model. She uses the midpoints of the classes to calculate estimates, \(m\) and \(s\), for the mean and standard deviation respectively, in centimetres, of the 150 heights. She then uses the distribution N\((m, s^2)\) as her model.
  1. Use Nina's model to find P\((20 < X < 30)\). [4]
    1. Complete the table in the Printed Answer Booklet to show the probabilities obtained from Sam's model and Nina's model. [2]
    2. By considering the different ranges of values of \(X\) given in the table, discuss how well the two models fit the original distribution. [2]
The heights, in centimetres, of a random sample of 150 plants of a certain variety were measured. The results are summarised in the histogram.

\includegraphics{figure_2}

One of the 150 plants is chosen at random, and its height, $X$ cm, is noted.

\begin{enumerate}[label=(\alph*)]
\item Show that P$(20 < X < 30) = 0.147$, correct to 3 significant figures. [2]
\end{enumerate}

Sam suggests that the distribution of $X$ can be well modelled by the distribution N$(40, 100)$.

\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item \begin{enumerate}[label=(\roman*)]
\item Give a brief justification for the use of the normal distribution in this context. [1]
\item Give a brief justification for the choice of the parameter values 40 and 100. [2]
\end{enumerate}

\item Use Sam's model to find P$(20 < X < 30)$. [1]
\end{enumerate}

Nina suggests a different model. She uses the midpoints of the classes to calculate estimates, $m$ and $s$, for the mean and standard deviation respectively, in centimetres, of the 150 heights. She then uses the distribution N$(m, s^2)$ as her model.

\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item Use Nina's model to find P$(20 < X < 30)$. [4]

\item \begin{enumerate}[label=(\roman*)]
\item Complete the table in the Printed Answer Booklet to show the probabilities obtained from Sam's model and Nina's model. [2]
\item By considering the different ranges of values of $X$ given in the table, discuss how well the two models fit the original distribution. [2]
\end{enumerate}
\end{enumerate}

\hfill \mbox{\textit{SPS SPS SM Statistics 2024 Q2 [14]}}
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