5.02c Linear coding: effects on mean and variance

7 James throws a discus repeatedly in an attempt to achieve a successful throw. A throw is counted as successful if the distance achieved is over 40 metres. For each throw, the probability that James is successful is \(\frac { 1 } { 4 }\), independently of all other throws. Find the probability that James takes

1. exactly 5 throws to achieve the first successful throw,
2. more than 8 throws to achieve the first successful throw. In order to qualify for a competition, a discus-thrower must throw over 40 metres within at most six attempts. When a successful throw is achieved, no further throws are taken. Find the probability that James qualifies for the competition. Colin is another discus-thrower. For each throw, the probability that he will achieve a throw over 40 metres is \(\frac { 1 } { 3 }\), independently of all other throws. Find the probability that exactly one of James and Colin qualifies for the competition.

5 Jasmine has one \(\\) 5\( coin, two \)\\( 2\) coins and two \(\\) 1\( coins. She selects two of these coins at random. The random variable \)X$ is the total value, in dollars, of these two coins.

1. Show that \(\mathrm { P } ( X = 7 ) = 0.2\).
2. Draw up the probability distribution table for \(X\).
3. Find the value of \(\operatorname { Var } ( X )\).

4 The 1300 train from Jahor to Keman runs every day. The probability that the train arrives late in Keman is 0.35 .

1. For a random sample of 7 days, find the probability that the train arrives late on fewer than 3 days.
   A random sample of 142 days is taken.
2. Use an approximation to find the probability that the train arrives late on more than 40 days.

7 The times, in minutes, that Karli spends each day on social media are normally distributed with mean 125 and standard deviation 24.

1. 1. On how many days of the year ( 365 days) would you expect Karli to spend more than 142 minutes on social media?
   2. Find the probability that Karli spends more than 142 minutes on social media on fewer than 2 of 10 randomly chosen days.
2. On \(90 \%\) of days, Karli spends more than \(t\) minutes on social media. Find the value of \(t\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

6 In a game, Jim throws three darts at a board. This is called a 'turn'. The centre of the board is called the bull's-eye. The random variable \(X\) is the number of darts in a turn that hit the bull's-eye. The probability distribution of \(X\) is given in the following table. It is given that \(\mathrm { E } ( X ) = 0.55\).

1. Find the values of \(p\) and \(q\).
2. Find \(\operatorname { Var } ( X )\).
   Jim is practising for a competition and he repeatedly throws three darts at the board.
3. Find the probability that \(X = 1\) in at least 3 of 12 randomly chosen turns.
4. Find the probability that Jim first succeeds in hitting the bull's-eye with all three darts on his 9th turn.

2 The residents of Persham were surveyed about the reliability of their internet service. 12\% rated the service as 'poor', \(36 \%\) rated it as 'satisfactory' and \(52 \%\) rated it as 'good'. A random sample of 8 residents of Persham is chosen.

1. Find the probability that more than 2 and fewer than 8 of them rate their internet service as poor or satisfactory.
   A random sample of 125 residents of Persham is now chosen.
2. Use an approximation to find the probability that more than 72 of these residents rate their internet service as good.

7 Sanket plays a game using a biased die which is twice as likely to land on an even number as on an odd number. The probabilities for the three even numbers are all equal and the probabilities for the three odd numbers are all equal.

1. Find the probability of throwing an odd number with this die. Sanket throws the die once and calculates his score by the following method.
   • If the number thrown is 3 or less he multiplies the number thrown by 3 and adds 1 .
   • If the number thrown is more than 3 he multiplies the number thrown by 2 and subtracts 4 .
   The random variable \(X\) is Sanket's score.
2. Show that \(\mathrm { P } ( X = 8 ) = \frac { 2 } { 9 }\). The table shows the probability distribution of \(X\).

   \(x\)	4	6	7	8	10
   \(\mathrm { P } ( X = x )\)	\(\frac { 3 } { 9 }\)	\(\frac { 1 } { 9 }\)	\(\frac { 2 } { 9 }\)	\(\frac { 2 } { 9 }\)	\(\frac { 1 } { 9 }\)

3. Given that \(\mathrm { E } ( X ) = \frac { 58 } { 9 }\), find \(\operatorname { Var } ( X )\). Sanket throws the die twice.
4. Find the probability that the total of the scores on the two throws is 16 .
5. Given that the total of the scores on the two throws is 16 , find the probability that the score on the first throw was 6 .

2 Esme noted the test marks, \(x\), of 16 people in a class. She found that \(\Sigma x = 824\) and that the standard deviation of \(x\) was 6.5.

1. Calculate \(\Sigma ( x - 50 )\) and \(\Sigma ( x - 50 ) ^ { 2 }\).
2. One person did the test later and her mark was 72. Calculate the new mean and standard deviation of the marks of all 17 people.

2 The discrete random variable \(X\) has the following probability distribution. Given that \(\mathrm { E } ( X ) = 2.3\) and \(\operatorname { Var } ( X ) = 3.01\), find the values of \(p , q\) and \(r\).

6 A fair spinner \(A\) has edges numbered \(1,2,3,3\). A fair spinner \(B\) has edges numbered \(- 3 , - 2 , - 1,1\). Each spinner is spun. The number on the edge that the spinner comes to rest on is noted. Let \(X\) be the sum of the numbers for the two spinners.

1. Copy and complete the table showing the possible values of \(X\).

   	Spinner \(A\)
   \cline { 2 - 6 }		1	2	3	3
   Spinner \(B\)	- 2
   - 2			1
   - 1
   1

2. Draw up a table showing the probability distribution of \(X\).
3. Find \(\operatorname { Var } ( X )\).
4. Find the probability that \(X\) is even, given that \(X\) is positive.

3 In the data-entry department of a certain firm, it is known that \(0.12 \%\) of data items are entered incorrectly, and that these errors occur randomly and independently.

1. A random sample of 3600 data items is chosen. The number of these data items that are incorrectly entered is denoted by \(X\).
   1. State the distribution of \(X\), including the values of any parameters.
   2. State an appropriate approximating distribution for \(X\), including the values of any parameters. Justify your choice of approximating distribution.
   3. Use your approximating distribution to find \(\mathrm { P } ( X > 2 )\).
2. Another large random sample of \(n\) data items is chosen. The probability that the sample contains no data items that are entered incorrectly is more than 0.1 . Use an approximating distribution to find the largest possible value of \(n\).

1 A fair coin is tossed 5 times and the number of heads is recorded.

1. The random variable \(X\) is the number of heads. State the mean and variance of \(X\).
2. The number of heads is doubled and denoted by the random variable \(Y\). State the mean and variance of \(Y\).

1 The mass, in kilograms, of a block of cheese sold in a supermarket is denoted by the random variable \(M\). The masses of a random sample of 40 blocks are summarised as follows. $$n = 40 \quad \Sigma m = 20.50 \quad \Sigma m ^ { 2 } = 10.7280$$

1. Calculate unbiased estimates of the population mean and variance of \(M\).
2. The price, \(\\) P\(, of a block of cheese of mass \)M \mathrm {~kg}\( is found using the formula \)P = 11 M + 0.50\(. Find estimates of the population mean and variance of \)P$.

4 The masses, in kilograms, of chemicals \(A\) and \(B\) produced per day by a factory are modelled by the independent random variables \(X\) and \(Y\) respectively, where \(X \sim \mathrm {~N} ( 10.3,5.76 )\) and \(Y \sim \mathrm {~N} ( 11.4,9.61 )\). The income generated by the chemicals is \(\\) 2.50\( per kilogram for \)A\( and \)\\( 3.25\) per kilogram for \(B\).

1. Find the mean and variance of the daily income generated by chemical \(A\). \includegraphics[max width=\textwidth, alt={}, center]{d42b3c4d-c426-4231-a35a-cac80dbdf82c-06_56_1566_495_333}
2. Find the probability that, on a randomly chosen day, the income generated by chemical \(A\) is greater than the income generated by chemical \(B\).

1 Exam marks, \(X\), have mean 70 and standard deviation 8.7. The marks need to be scaled using the formula \(Y = a X + b\) so that the scaled marks, \(Y\), have mean 55 and standard deviation 6.96. Find the values of \(a\) and \(b\).

5

1. Deng wishes to test whether a certain coin is biased so that it is more likely to show Heads than Tails. He throws it 12 times. If it shows Heads more than 9 times, he will conclude that the coin is biased. Calculate the significance level of the test.
2. Deng throws another coin 100 times in order to test, at the \(5 \%\) significance level, whether it is biased towards Heads. Find the rejection region for this test.

5 The probability that a new car of a certain type has faulty brakes is 0.008 . A random sample of 520 new cars of this type is chosen, and the number, \(X\), having faulty brakes is noted.

1. Describe fully the distribution of \(X\) and describe also a suitable approximating distribution. Justify this approximating distribution.
2. Use your approximating distribution to find
   1. \(\mathrm { P } ( X > 3 )\),
   2. the smallest value of \(n\) such that \(\mathrm { P } ( X = n ) > \mathrm { P } ( X = n + 1 )\).

7 Leila suspects that a particular six-sided die is biased so that the probability, \(p\), that it will show a six is greater than \(\frac { 1 } { 6 }\). She tests the die by throwing it 5 times. If it shows a six on 3 or more throws she will conclude that it is biased.

1. State what is meant by a Type I error in this situation and calculate the probability of a Type I error.
2. Assuming that the value of \(p\) is actually \(\frac { 2 } { 3 }\), calculate the probability of a Type II error. Leila now throws the die 80 times and it shows a six on 50 throws.
3. Calculate an approximate \(96 \%\) confidence interval for \(p\).

1 It is known that \(1.2 \%\) of rods made by a certain machine are bent. The random variable \(X\) denotes the number of bent rods in a random sample of 400 rods.

1. State the distribution of \(X\).
2. State, with a reason, a suitable approximate distribution for \(X\).
3. Use your approximate distribution to find the probability that the sample will include more than 2 bent rods.

2 An engineering test consists of 100 multiple-choice questions. Each question has 5 suggested answers, only one of which is correct. Ashok knows nothing about engineering, but he claims that his general knowledge enables him to get more questions correct than just by guessing. Ashok actually gets 27 answers correct. Use a suitable approximating distribution to test at the \(5 \%\) significance level whether his claim is justified.

1 The hourly wages, \(\pounds x\), of a random sample of 60 employees working for a company are summarised as follows. $$n = 60 \quad \sum x = 759.00 \quad \sum x ^ { 2 } = 11736.59$$

1. Calculate the mean and standard deviation of \(x\).
2. The workers are offered a wage increase of \(2 \%\). Use your answers to part (i) to deduce the new mean and standard deviation of the hourly wages after this increase.
3. As an alternative the workers are offered a wage increase of 25 p per hour. Write down the new mean and standard deviation of the hourly wages after this 25p increase.

6 A retail analyst records the numbers of loaves of bread of a particular type bought by a sample of shoppers in a supermarket.

1. Calculate the mean and standard deviation of the numbers of loaves bought per person.
2. Each loaf costs \(\pounds 1.04\). Calculate the mean and standard deviation of the amount spent on loaves per person.

2 Dwayne is a car salesman. The numbers of cars, \(x\), sold by Dwayne each month during the year 2008 are summarised by $$n = 12 , \quad \Sigma x = 126 , \quad \Sigma x ^ { 2 } = 1582 .$$

1. Calculate the mean and standard deviation of the monthly numbers of cars sold.
2. Dwayne earns \(\pounds 500\) each month plus \(\pounds 100\) commission for each car sold. Show that the mean of Dwayne's monthly earnings is \(\pounds 1550\). Find the standard deviation of Dwayne's monthly earnings.
3. Marlene is a car saleswoman and is paid in the same way as Dwayne. During 2008 her monthly earnings have mean \(\pounds 1625\) and standard deviation \(\pounds 280\). Briefly compare the monthly numbers of cars sold by Marlene and Dwayne during 2008.

5 A pottery manufacturer makes teapots in batches of 50. On average 3\% of teapots are faulty.

1. Find the probability that in a batch of 50 there is
   (A) exactly one faulty teapot,
   (B) more than one faulty teapot.
2. The manufacturer produces 240 batches of 50 teapots during one month. Find the expected number of batches which contain exactly one faulty teapot.

2 The marks \(x\) scored by a sample of 56 students in an examination are summarised by $$n = 56 , \quad \Sigma x = 3026 , \quad \Sigma x ^ { 2 } = 178890 .$$

1. Calculate the mean and standard deviation of the marks.
2. The highest mark scored by any of the 56 students in the examination was 93. Show that this result may be considered to be an outlier.
3. The formula \(y = 1.2 x - 10\) is used to scale the marks. Find the mean and standard deviation of the scaled marks.