Estimate from summary statistics

4

1. Amy measured her pulse rate while resting, \(x\) beats per minute, at the same time each day on 30 days. The results are summarised below. $$\Sigma ( x - 80 ) = - 147 \quad \Sigma ( x - 80 ) ^ { 2 } = 952$$ Find the mean and standard deviation of Amy's pulse rate.
2. Amy's friend Marok measured her pulse rate every day after running for half an hour. Marok's pulse rate, in beats per minute, was found to have a mean of 148.6 and a standard deviation of 18.5. Assuming that pulse rates have a normal distribution, find what proportion of Marok's pulse rates, after running for half an hour, were above 160 beats per minute.

4 The random variable \(Y\) has the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). The results of 40 independent observations of \(Y\) are summarised by $$\Sigma y = 3296.0 , \quad \Sigma y ^ { 2 } = 286800.40$$

1. Calculate unbiased estimates of \(\mu\) and \(\sigma ^ { 2 }\).
2. Use your answers to part (i) to estimate the probability that a single random observation of \(Y\) will be less than 60.0.
3. Explain whether it is necessary to know that \(Y\) is normally distributed in answering part (i) of this question.

2 A random variable \(C\) has the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). A random sample of 10 observations of \(C\) is obtained, and the results are summarised as $$n = 10 , \Sigma c = 380 , \Sigma c ^ { 2 } = 14602 .$$

1. Calculate unbiased estimates of \(\mu\) and \(\sigma ^ { 2 }\).
2. Hence calculate an estimate of the probability that \(C > 40\).

13 In the South West region of England, 100 households were randomly selected and, for each household, the weekly expenditure, \(\pounds X\), per person on food and drink was recorded. The maximum amount recorded was \(\pounds 40.48\) and the minimum amount recorded was £22.00 The results are summarised below, where \(\bar { X }\) denotes the sample mean. $$\sum x = 3046.14 \quad \sum ( x - \bar { x } ) ^ { 2 } = 1746.29$$ 13

1. 1. Find the mean of \(X\)
      Find the standard deviation of \(X\)
      [0pt] [2 marks] 13
2. (ii) Using your results from part (a)(i) and other information given, explain why the normal distribution can be used to model \(X\).
   [0pt] [2 marks] 13
3. (iii) Find the probability that a household in the South West spends less than \(\pounds 25.00\) on food and drink per person per week.
   13
4. For households in the North West of England, the weekly expenditure, \(\pounds Y\), per person on food and drink can be modelled by a normal distribution with mean \(\pounds 29.55\) It is known that \(\mathrm { P } ( Y < 30 ) = 0.55\)
   Find the standard deviation of \(Y\), giving your answer to one decimal place.
   [0pt] [3 marks]

9 At the beginning of the academic year, all the pupils in year 12 at a college take part in an assessment. Summary statistics for the marks obtained by the 2021 cohort are given below.
\(n = 205 \sum x = 23042 \sum x ^ { 2 } = 2591716\) Marks may only be whole numbers, but the Head of Mathematics believes that the distribution of marks may be modelled by a Normal distribution.

1. Calculate
   • The mean mark
   • The variance of the marks
   • Use your answers to part (a) to write down a possible Normal model for the distribution of marks.
   One candidate in the cohort scored less than 105.
2. Determine whether the model found in part (b) is consistent with this information.
3. Use the model to calculate an estimate of the number of candidates who scored 115 marks.

18 Riley is investigating the daily water consumption, in litres, of his household.
He records the amount used for a random sample of 120 days from the previous twelve-month period. The daily water consumption, in litres, is denoted by \(x\). Summary statistics for Riley's sample are given below.
\(\sum \mathrm { x } = 31164.7 \sum \mathrm { x } ^ { 2 } = 8101050.91 \mathrm { n } = 120\)

1. Calculate the sample mean giving your answer correct to \(\mathbf { 3 }\) significant figures. Riley displays the data in a histogram.
   \includegraphics[max width=\textwidth, alt={}, center]{11788aaf-98fb-4a78-8a40-a40743b1fe15-13_832_1383_934_242}
2. Find the number of days on which between 255 and 260 litres were used.
3. Give two reasons why a Normal distribution may be an appropriate model for the daily consumption of water. Riley uses the sample mean and the sample variance, both correct to \(\mathbf { 3 }\) significant figures, as parameters of a Normal distribution to model the daily consumption of water.
4. Use Riley's model to calculate the probability that on a randomly chosen day the household uses less than 255 litres of water.
5. Calculate the probability that the household uses less than 255 litres of water on at least 5 days out of a random sample of 28 days. The company which supplies the water makes charges relating to water consumption which are shown in the table below.

   Standing charge per day in pence	7.8
   Charge per litre in pence	0.18

6. Adapt Riley's model for daily water consumption to model the daily charges for water consumption. \section*{END OF QUESTION PAPER}

1. At the beginning of the academic year, all the pupils in year 12 at a college take part in an assessment. Summary statistics for the marks obtained by the 2021 cohort are given below.
\(n = 205 \quad \sum x = 23042 \quad \sum x ^ { 2 } = 2591716\) Marks may only be whole numbers, but the Head of Mathematics believes that the distribution of marks may be modelled by a Normal distribution.

1. Calculate
   • The mean mark
   • The variance of the marks
   • Use your answers to part (a) to write down a possible Normal model for the distribution of marks.