Approximating Binomial to Normal Distribution

3 On a certain road \(20 \%\) of the vehicles are trucks, \(16 \%\) are buses and the remainder are cars.

1. A random sample of 11 vehicles is taken. Find the probability that fewer than 3 are buses.
2. A random sample of 125 vehicles is now taken. Using a suitable approximation, find the probability that more than 73 are cars.

3 A fair dice is thrown 90 times. Use an appropriate approximation to find the probability that the number 1 is obtained 14 or more times.

1. The probability that a person completes a particular task in less than 15 minutes is 0.4 Jeffrey selects 20 people at random and asks them to complete the task. The random variable, \(X\), represents the number of people who complete the task in less than 15 minutes.
   1. Find \(\mathrm { P } ( 5 \leqslant X < 8 )\)
   Mia takes a random sample of 140 people.
   Using a normal approximation, the probability that fewer than \(n\) of these 140 people complete the task in less than 15 minutes is 0.0239 to 4 decimal places.
2. Find the value of \(n\) Show your working clearly.

13 It is known that \(26 \%\) of adults in the UK use a certain app. A researcher selects a random sample of 5000 adults in the UK. The random variable \(X\) is defined as the number of adults in the sample who use the app. Given that \(\mathrm { P } ( X < n ) < 0.025\), calculate the largest possible value of \(n\).

3 The random variable \(F\) has the distribution \(\mathrm { B } ( 40,0.65 )\). Use a suitable approximation to find \(\mathrm { P } ( F \leqslant 30 )\), justifying your approximation.

1. The random variable \(Y \sim \mathrm {~B} ( n , p )\).

Using a normal approximation the probability that \(Y\) is at least 65 is 0.2266 and the probability that \(Y\) is more than 52 is 0.8944 Find the value of \(n\) and the value of \(p\).

3. A die is rolled 60 times, and results in 16 sixes.

1. Use a suitable approximation to test, at the \(5 \%\) significance level, whether the probability of scoring a six is \(\frac { 1 } { 6 }\) or not. State your hypotheses clearly.
2. Describe how you would change the test if you wished to investigate whether the probability of scoring a six is greater than \(\frac { 1 } { 6 }\). Carry out this modified test.

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.

3 Tennis balls are dropped from a standard height, and the height of bounce, \(H \mathrm {~cm}\), is measured. \(H\) is a random variable with the distribution \(\mathrm { N } \left( 40 , \sigma ^ { 2 } \right)\). It is given that \(\mathrm { P } ( H < 32 ) = 0.2\).

1. Find the value of \(\sigma\).
2. 90 tennis balls are selected at random. Use an appropriate approximation to find the probability that more than 19 have \(H < 32\).

2 An airline has found that, on average, 1 in 100 passengers do not arrive for each flight, and that this occurs randomly. For one particular flight the airline always sells 403 seats. The plane only has room for 400 passengers, so the flight is overbooked if the number of passengers who do not arrive is less than 3 . Use a suitable approximation to find the probability that the flight is overbooked.

1. Xian rolls a fair die 10 times.

The random variable \(X\) represents the number of times the die lands on a six.

1. Using a suitable distribution for \(X\), find
   1. \(\mathrm { P } ( X = 3 )\)
   2. \(\mathrm { P } ( X < 3 )\) Xian repeats this experiment each day for 60 days and records the number of days when \(X = 3\)
2. Find the probability that there were at least 12 days when \(X = 3\)
3. Find an estimate for the total number of sixes that Xian will roll during these 60 days.
4. Use a normal approximation to estimate the probability that Xian rolls a total of more than 95 sixes during these 60 days.

5 The random variable \(W\) has the distribution \(\mathbf { B } ( 30 , p )\).

1. Use the exact binomial distribution to calculate \(\mathbf { P } ( W = 10 )\) when \(p = 0.4\).
2. Find the range of values of \(p\) for which you would expect that a normal distribution could be used as an approximation to the distribution of \(W\).
3. Use a normal approximation to calculate \(\mathrm { P } ( W = 10 )\) when \(p = 0.4\).

5 A fair six sid dl e,w itlf aces mark dress s ther im imes.

1. Use ara \(p\) in matin of id b pb b lity b ta 3 s ob ain of ewer th rㅇs imes. [4]
2. Js tifys se 6 th ap ox matin pe rt (a). Ora \(\mathbf { h }\) b roccasity he same \(\dot { \mathbf { d } }\) e is th \(\boldsymbol { w }\) ep ated y il a \(\mathbf { 3 } \mathrm { sb }\) aie d
3. Fid b pb b lity b tb ain g ʒ eq res fewer th \(n\) st \(h\) s.

5 On trains in the morning rush hour, each person is either a student with probability 0.36 , or an office worker with probability 0.22 , or a shop assistant with probability 0.29 or none of these.

1. 8 people on a morning rush hour train are chosen at random. Find the probability that between 4 and 6 inclusive are office workers.
2. 300 people on a morning rush hour train are chosen at random. Find the probability that between 31 and 49 inclusive are neither students nor office workers nor shop assistants.

4 DVD players are tested after manufacture. The probability that a randomly chosen DVD player is defective is 0.02 . The number of defective players in a random sample of size 80 is denoted by \(R\).

1. Use an appropriate approximation to find \(\mathrm { P } ( R \geqslant 2 )\).
2. Find the smallest value of \(r\) for which \(\mathrm { P } ( R \geqslant r ) < 0.01\).

5 The random variable \(X\) has a binomial distribution with parameters \(n\) and \(p\).

1. Given that $$\mathrm { E } ( X ) = n p \quad \text { and } \quad \mathrm { E } ( X ( X - 1 ) ) = n ( n - 1 ) p ^ { 2 }$$ find an expression for \(\operatorname { Var } ( X )\).
2. Given that \(X\) has a mean of 36 and a standard deviation of 4.8:
   1. find values for \(n\) and \(p\);
   2. use a distributional approximation to estimate \(\mathrm { P } ( 30 < X < 40 )\).

2 On average, 2 apples out of 15 are classified as being underweight. Find the probability that in a random sample of 200 apples, the number of apples which are underweight is more than 21 and less than 35.

3 In a random sample of 100 students at Luciana's college, \(x\) students said that they liked exams. Luciana used this result to find an approximate \(90 \%\) confidence interval for the proportion, \(p\), of all students at her college who liked exams. Her confidence interval had width 0.15792 .

1. Find the two possible values of \(x\).
   Suzma independently took another random sample and found another approximate \(90 \%\) confidence interval for \(p\).
2. Find the probability that neither of the two confidence intervals contains the true value of \(p\). [1]

4 A continuous random variable is normally distributed with mean \(\mu\). A significance test for \(\mu\) is carried out, at the \(5 \%\) significance level, on 90 independent occasions.

1. Given that the null hypothesis is correct on all 90 occasions, use a suitable approximation to find the probability that on 6 or fewer occasions the test results in a Type I error. Justify your approximation.
2. Given instead that on all 90 occasions the probability of a Type II error is 0.35 , use a suitable approximation to find the probability that on fewer than 29 occasions the test results in a Type II error.

2 A village has a population of 600 people. A sample of 12 people is obtained as follows. A list of all 600 people is obtained and a three-digit number, between 001 and 600 inclusive, is allocated to each name in alphabetical order. Twelve three-digit random numbers, between 001 and 600 inclusive, are obtained and the people whose names correspond to those numbers are chosen.

1. Find the probability that all 12 of the numbers chosen are 500 or less.
2. When the selection has been made, it is found that all of the numbers chosen are 500 or less. One of the people in the village says, "The sampling method must have been biased." Comment on this statement.