OCR H240/02 2018 September — Question 9 12 marks

Exam BoardOCR
ModuleH240/02 (Pure Mathematics and Statistics)
Year2018
SessionSeptember
Marks12
TopicNormal Distribution
TypeEstimate from grouped frequency data
DifficultyModerate -0.3 This is a standard Stats 1 question combining histogram interpretation with normal distribution application. Parts (i)-(iii) involve routine histogram reading and grouped data calculations, while (iv) requires a straightforward normal probability calculation. Part (v) asks for a qualitative comment using the 95% rule rather than detailed analysis. All techniques are textbook exercises with no novel problem-solving required, making it slightly easier than average.
Spec2.02a Interpret single variable data: tables and diagrams2.02b Histogram: area represents frequency2.02f Measures of average and spread2.02g Calculate mean and standard deviation2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation
9 The finance department of a retail firm recorded the daily income each day for 300 days. The results are summarised in the histogram. \includegraphics[max width=\textwidth, alt={}, center]{85de9a39-f8be-40ee-b0c8-e2e632be93d8-6_689_1575_488_246}
  1. Find the number of days on which the daily income was between \(\pounds 4000\) and \(\pounds 6000\).
  2. Calculate an estimate of the number of days on which the daily income was between \(\pounds 2700\) and \(\pounds 3600\).
  3. Use the midpoints of the classes to show that an estimate of the mean daily income is \(\pounds 3275\). An estimate of the standard deviation of the daily income is \(\pounds 1060\). The finance department uses the distribution \(\mathrm { N } \left( 3275,1060 ^ { 2 } \right)\) to model the daily income, in pounds.
  4. Calculate the number of days on which, according to this model, the daily income would be between \(\pounds 4000\) and \(\pounds 6000\).
  5. It is given that approximately \(95 \%\) of values of the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\) lie within the range \(\mu \pm 2 \sigma\). Without further calculation, use this fact to comment briefly on whether the proposed model is a good fit to the data illustrated in the histogram.
9 The finance department of a retail firm recorded the daily income each day for 300 days. The results are summarised in the histogram.\\
\includegraphics[max width=\textwidth, alt={}, center]{85de9a39-f8be-40ee-b0c8-e2e632be93d8-6_689_1575_488_246}\\
(i) Find the number of days on which the daily income was between $\pounds 4000$ and $\pounds 6000$.\\
(ii) Calculate an estimate of the number of days on which the daily income was between $\pounds 2700$ and $\pounds 3600$.\\
(iii) Use the midpoints of the classes to show that an estimate of the mean daily income is $\pounds 3275$.

An estimate of the standard deviation of the daily income is $\pounds 1060$. The finance department uses the distribution $\mathrm { N } \left( 3275,1060 ^ { 2 } \right)$ to model the daily income, in pounds.\\
(iv) Calculate the number of days on which, according to this model, the daily income would be between $\pounds 4000$ and $\pounds 6000$.\\
(v) It is given that approximately $95 \%$ of values of the distribution $\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)$ lie within the range $\mu \pm 2 \sigma$. Without further calculation, use this fact to comment briefly on whether the proposed model is a good fit to the data illustrated in the histogram.

\hfill \mbox{\textit{OCR H240/02 2018 Q9 [12]}}
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