5.02c Linear coding: effects on mean and variance

6 The temperature of a supermarket fridge is regularly checked to ensure that it is working correctly. Over a period of three months the temperature (measured in degrees Celsius) is checked 600 times. These temperatures are displayed in the cumulative frequency diagram below. \includegraphics[max width=\textwidth, alt={}, center]{7b92607f-1bf9-45f0-997b-fe76c88b5fcd-4_1054_1649_539_248}

1. Use the diagram to estimate the median and interquartile range of the data.
2. Use your answers to part (i) to show that there are very few, if any, outliers in the sample.
3. Suppose that an outlier is identified in these data. Discuss whether it should be excluded from any further analysis.
4. Copy and complete the frequency table below for these data.

   Temperature \(( t\) degrees Celsius \()\)	\(3.0 \leqslant t \leqslant 3.4\)	\(3.4 < t \leqslant 3.8\)	\(3.8 < t \leqslant 4.2\)	\(4.2 < t \leqslant 4.6\)	\(4.6 < t \leqslant 5.0\)
   Frequency			243	157

5. Use your table to calculate an estimate of the mean.
6. The standard deviation of the temperatures in degrees Celsius is 0.379 . The temperatures are converted from degrees Celsius into degrees Fahrenheit using the formula \(F = 1.8 C + 32\). Hence estimate the mean and find the standard deviation of the temperatures in degrees Fahrenheit.

4 The probability distribution of the random variable \(X\) is given by the formula $$\mathrm { P } ( X = r ) = \frac { k } { r ( r - 1 ) } \text { for } r = 2,3,4,5,6 .$$

1. Show that the value of \(k\) is 1.2 . Using this value of \(k\), show the probability distribution of \(X\) in a table.
2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

4

1. In Germany, towards the end of the nineteenth century, a study was undertaken into the distribution of the sexes in families of various sizes. The table shows some data about the numbers of girls in 500 families, each with 5 children. It is thought that the binomial distribution \(\mathrm { B } ( 5 , p )\) should model these data.

   Number of girls	Number of families
   0	32
   1	110
   2	154
   3	125
   4	63
   5	16

   1. Use this information to calculate an estimate for the mean number of girls per family of 5 children. Hence show that 0.45 can be taken as an estimate of \(p\).
   2. Investigate at a \(5 \%\) significance level whether the binomial model with \(p\) estimated as 0.45 fits the data. Comment on your findings and also on the extent to which the conditions for a binomial model are likely to be met.
2. A researcher wishes to select 50 families from the 500 in part (a) for further study. Suggest what sort of sample she might choose and describe how she should go about choosing it.

14 The inequality $$\left( x ^ { 2 } - 5 x - 24 \right) \left( x ^ { 2 } + 7 x + a \right) < 0$$ has the solution set $$\{ x : - 9 < x < - 3 \} \cup \{ x : 2 < x < b \}$$ Find the values of integers \(a\) and \(b\) \includegraphics[max width=\textwidth, alt={}]{b37e2ee7-1cde-4d75-895a-381b32f4e95a-21_2491_1755_173_123} number Additional page, if required. Write the question numbers in the left-hand margin. \(\_\_\_\_\) number \section*{Additional page, if required. Write the question numbers in the left-hand margin.
Additional page, if required. uestion numbers in the left-hand margin.}

1 The discrete random variable \(X\) has \(\operatorname { Var } ( X ) = 6.5\) Find \(\operatorname { Var } ( 4 X - 2 )\) Circle your answer.
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3 The discrete random variable \(R\) has the following probability distribution. It is known that \(\mathrm { E } ( R ) = 0.2\) and \(\operatorname { Var } ( R ) = 3.56\) Find the values of \(a , b\) and \(c\).
[0pt] [4 marks]

1 The discrete random variable \(A\) has the following probability distribution.

1. Find the value of \(\mathrm { E } ( A )\).
2. Determine the value of \(\operatorname { Var } ( A )\).
3. The variable \(A\) represents the value in pence of a coin chosen at random from a pile. Mia picks one coin at random from the pile. She then adds, from a different source, another coin of the same value as the one that she has chosen, and one 50p coin.
   1. Find the mean of the value of the three coins.
   2. Find the variance of the value of the three coins.

1 A discrete random variable \(X\) has the following distribution, where \(a , b\) and \(c\) are constants. It is given that \(\mathrm { E } ( X ) = 1.25\) and \(\operatorname { Var } ( X ) = 0.8875\).

1. Determine the values of \(a\), \(b\) and \(c\).
2. The random variable \(Y\) is defined by \(Y = 7 - 2 X\). Write down the value of \(\operatorname { Var } ( Y )\).
3. Twenty independent observations of \(X\) are obtained. The number of those observations for which \(X = 3\) is denoted by \(T\). Find the value of \(\operatorname { Var } ( T )\).

2 A discrete random variable \(D\) has the following probability distribution, where \(a\) is a constant. Determine the value of \(\operatorname { Var } ( 3 D + 4 )\).

1. The discrete random variable \(X\) has the following probability distribution

Given that \(\mathrm { E } ( X ) = 0.5\)

1. find the value of \(a\). Given also that \(\operatorname { Var } ( X ) = 5.01\)
2. find the value of \(b\) and the value of \(c\). The random variable \(Y = 5 - 8 X\)
3. Find
   1. \(\mathrm { E } ( Y )\)
   2. \(\operatorname { Var } ( Y )\)
4. Find \(\mathrm { P } \left( 4 X ^ { 2 } > Y \right)\)

1. A red spinner is designed so that the score \(R\) is given by the following probability distribution.

1. Show that \(\mathrm { E } \left( R ^ { 2 } \right) = 15.8\) Given also that \(\mathrm { E } ( R ) = 3.7\)
2. find the standard deviation of \(R\), giving your answer to 2 decimal places. A yellow spinner is designed so that the score \(Y\) is given by the probability distribution in the table below. The cumulative distribution function \(\mathrm { F } ( y )\) is also given.

   \(y\)	2	3	4	5	6
   \(\mathrm { P } ( Y = y )\)	0.1	0.2	0.1	\(a\)	\(b\)
   \(\mathrm {~F} ( y )\)	0.1	0.3	0.4	\(c\)	\(d\)

3. Write down the value of \(d\) Given that \(\mathrm { E } ( Y ) = 4.55\)
4. find the value of \(c\) Pabel and Jessie play a game with these two spinners.
   Pabel uses the red spinner.
   Jessie uses the yellow spinner.
   They take turns to spin their spinner.
   The winner is the first person whose spinner lands on the number 2 and the game ends. Jessie spins her spinner first.
5. Find the probability that Jessie wins on her second spin.
6. Calculate the probability that, in a game, the score on Pabel's first spin is the same as the score on Jessie's first spin.

2. A spinner can land on the numbers \(2,4,5,7\) or 8 only. The random variable \(X\) represents the number that this spinner lands on when it is spun once. The probability distribution of \(X\) is given in the table below.

1. Find \(\mathrm { P } ( 2 X - 3 > 5 )\) Given that \(\mathrm { E } ( X ) = 4.6\)
2. show that \(\operatorname { Var } ( X ) = 4.14\) The random variable \(Y = a X - b\) where \(a\) and \(b\) are positive constants.
   Given that $$\mathrm { E } ( Y ) = 13.4 \quad \text { and } \quad \operatorname { Var } ( Y ) = 66.24$$
3. find the value of \(a\) and the value of \(b\) In a game Sam and Alex each spin the spinner once, landing on \(X _ { 1 }\) and \(X _ { 2 }\) respectively.
   Sam's score is given by the random variable \(S = X _ { 1 }\) Alex's score is given by the random variable \(R = 2 X _ { 2 } - 3\) The person with the higher score wins the game. If the scores are the same it is a draw.
4. Find the probability that Sam wins the game.

1. The discrete random variable \(X\) has probability distribution

where \(a\) and \(b\) are constants.

1. Write down an equation for \(a\) and \(b\).
2. Calculate \(\mathrm { E } ( X )\). Given that \(\mathrm { E } \left( X ^ { 2 } \right) = 3.5\)
3. 1. find a second equation in \(a\) and \(b\),
   2. hence find the value of \(a\) and the value of \(b\).
4. Find \(\operatorname { Var } ( X )\). The random variable \(Y = 5 - 3 X\)
5. Find \(\mathrm { P } ( Y > 0 )\).
6. Find
   1. \(\mathrm { E } ( Y )\),
   2. \(\operatorname { Var } ( Y )\).

1. The discrete random variable \(X\) is defined by the cumulative distribution function

where \(k\) is a constant.

1. Find the probability distribution of \(X\).
2. Find \(\mathrm { P } ( 1.5 < X \leqslant 3.5 )\) The random variable \(Y = 12 - 7 X\)
3. Calculate Var(Y)
4. Calculate \(\mathrm { P } ( 4 X \leqslant | Y | )\)

1. Morgan is investigating the body length, \(b\) centimetres, of squirrels.

A random sample of 8 squirrels is taken and the data for each squirrel is coded using $$x = \frac { b - 21 } { 2 }$$ The results for the coded data are summarised below $$\sum x = - 1.2 \quad \sum x ^ { 2 } = 5.1$$

1. Find the mean of \(b\)
2. Find the standard deviation of \(b\) A 9th squirrel is added to the sample. Given that for all 9 squirrels \(\sum x = 0\)
3. find
   1. the body length of the 9th squirrel,
   2. the standard deviation of \(x\) for all 9 squirrels.

1. Adana selects one number at random from the distribution of \(X\) which has the following probability distribution.

1. Given that the number selected by Adana is not 5 , write down the probability it is 0
2. Show that \(\mathrm { E } \left( X ^ { 2 } \right) = 75\)
3. Find \(\operatorname { Var } ( X )\)
4. Find \(\operatorname { Var } ( 4 - 3 X )\) Bruno and Charlie each independently select one number at random from the distribution of \(X\)
5. Find the probability that the number Bruno selects is greater than the number Charlie selects. Devika multiplies Bruno's number by Charlie's number to obtain a product, \(D\)
6. Determine the probability distribution of \(D\)

1. The discrete random variable \(X\) has the following probability distribution.

1. Show that \(\mathrm { E } \left( \frac { 1 } { X } \right) = \frac { 2 } { 5 }\)
2. Find \(\operatorname { Var } \left( \frac { 1 } { X } \right)\) The random variable \(Y = \frac { 30 } { X }\)
3. Find
   1. \(\mathrm { E } ( Y )\)
   2. \(\operatorname { Var } ( Y )\)
4. Find \(\mathrm { P } ( X < 3 \mid Y < 20 )\)

1. A researcher recorded the time, \(t\) minutes, spent using a mobile phone during a particular afternoon, for each child in a club.

The researcher coded the data using \(v = \frac { t - 5 } { 10 }\) and the results are summarised in the table below. $$\text { (You may use } \sum f m = 825 \text { and } \sum f m ^ { 2 } = 12012.5 \text { ) }$$

1. Write down the value of \(a\) and the value of \(b\).
2. Calculate an estimate of the mean of \(v\).
3. Calculate an estimate of the standard deviation of \(v\).
4. Use linear interpolation to estimate the median of \(v\).
5. Hence describe the skewness of the distribution. Give a reason for your answer.
6. Calculate estimates of the mean and the standard deviation of the time spent using a mobile phone during the afternoon by the children in this club. \(\_\_\_\_\) VAYV SIHI NI JIIIM ION OC
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1. The discrete random variable \(X\) has probability distribution given by

where \(a\) is a constant.

1. Find the value of \(a\).
2. Write down \(\mathrm { E } ( X )\).
3. Find \(\operatorname { Var } ( X )\). The random variable \(Y = 6 - 2 X\)
4. Find \(\operatorname { Var } ( Y )\).
5. Calculate \(\mathrm { P } ( X \geqslant Y )\).

2. The random variable \(X\) has probability distribution

1. Given that \(\mathrm { E } ( X ) = 3.5\), write down two equations involving \(p\) and \(q\). Find
2. the value of \(p\) and the value of \(q\),
3. \(\operatorname { Var } ( X )\),
4. \(\operatorname { Var } ( 3 - 2 X )\).

1. The random variable \(X\) has probability function

$$\mathrm { P } ( X = x ) = \frac { ( 2 x - 1 ) } { 36 } \quad x = 1,2,3,4,5,6$$

1. Construct a table giving the probability distribution of \(X\). Find
2. \(\mathrm { P } ( 2 < X \leqslant 5 )\),
3. the exact value of \(\mathrm { E } ( X )\).
4. Show that \(\operatorname { Var } ( X ) = 1.97\) to 3 significant figures.
5. Find \(\operatorname { Var } ( 2 - 3 X )\).

1. The discrete random variable \(X\) has the probability distribution

1. Show that \(k = 0.1\) Find
2. \(\mathrm { E } ( X )\)
3. \(\mathrm { E } \left( X ^ { 2 } \right)\)
4. \(\operatorname { Var } ( 2 - 5 X )\) Two independent observations \(X _ { 1 }\) and \(X _ { 2 }\) are made of \(X\).
5. Show that \(\mathrm { P } \left( X _ { 1 } + X _ { 2 } = 4 \right) = 0.1\)
6. Complete the probability distribution table for \(X _ { 1 } + X _ { 2 }\)

   \(y\)	2	3	4	5	6	7	8
   \(\mathrm { P } \left( X _ { 1 } + X _ { 2 } = y \right)\)	0.01	0.04	0.10		0.25	0.24

7. Find \(\mathrm { P } \left( 1.5 < X _ { 1 } + X _ { 2 } \leqslant 3.5 \right)\)

3. The discrete random variable \(X\) can take only the values \(2,3,4\) or 6 . For these values the probability distribution function is given by where \(k\) is a positive integer.

1. Show that \(k = 3\) Find
2. \(\mathrm { F } ( 3 )\)
3. \(\mathrm { E } ( X )\)
4. \(\mathrm { E } \left( X ^ { 2 } \right)\)
5. \(\operatorname { Var } ( 7 X - 5 )\)

2. The discrete random variable \(X\) can take only the values 1,2 and 3 . For these values the cumulative distribution function is defined by $$\mathrm { F } ( x ) = \frac { x ^ { 3 } + k } { 40 } \quad x = 1,2,3$$

1. Show that \(k = 13\)
2. Find the probability distribution of \(X\). Given that \(\operatorname { Var } ( X ) = \frac { 259 } { 320 }\)
3. find the exact value of \(\operatorname { Var } ( 4 X - 5 )\).

4. The discrete random variable \(X\) has the probability function shown in the table below. Find

1. \(\alpha\),
2. \(\mathrm { P } ( - 1 < X \leq 2 )\),
3. \(\mathrm { F } ( - 0.4 )\),
4. \(\mathrm { E } ( 3 X + 4 )\),
5. \(\operatorname { Var } ( 2 X + 3 )\).