Find parameter from normal approximation

9

1. The random variable \(G\) has the distribution \(\mathrm { B } ( n , 0.75 )\). Find the set of values of \(n\) for which the distribution of \(G\) can be well approximated by a normal distribution.
2. The random variable \(H\) has the distribution \(\mathrm { B } ( n , p )\). It is given that, using a normal approximation, \(\mathrm { P } ( H \geqslant 71 ) = 0.0401\) and \(\mathrm { P } ( H \leqslant 46 ) = 0.0122\).
   1. Find the mean and standard deviation of the approximating normal distribution.
   2. Hence find the values of \(n\) and \(p\). 4

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 \(p\).

1. The random variable \(Y \sim \mathrm {~B} ( 225 , p )\)

Using a normal approximation, the probability that \(Y\) is at least 188 is 0.1056 to 4 decimal places.

1. Show that \(p\) satisfies \(145 p ^ { 2 } - 241 p + 100 = 0\) when the normal probability tables are used.
2. Hence find the value of \(p\), justifying your answer.

6. A fair 6 -sided die is thrown \(n\) times. The number of sixes, \(X\), is recorded. Using a normal approximation, \(\mathrm { P } ( X < 50 ) = 0.0082\) correct to 4 decimal places. Find the value of \(n\).
(10)

5. In a large school, \(20 \%\) of students own a touch screen laptop. A random sample of \(n\) students is chosen from the school. Using a normal approximation, the probability that more than 55 of these \(n\) students own a touch screen laptop is 0.0401 correct to 3 significant figures. Find the value of \(n\).
(8)
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5. The time taken for a randomly selected person to complete a test is \(M\) minutes, where \(M \sim \mathrm {~N} \left( 14 , \sigma ^ { 2 } \right)\) Given that \(10 \%\) of people take less than 12 minutes to complete the test,

1. find the value of \(\sigma\) Graham selects 15 people at random.
2. Find the probability that fewer than 2 of these people will take less than 12 minutes to complete the test. Jovanna takes a random sample of \(n\) people. Using a normal approximation, the probability that fewer than 9 of these \(n\) people will take less than 12 minutes to complete the test is 0.3085 to 4 decimal places.
3. Find the value of \(n\).

3. For a particular type of bulb, \(36 \%\) grow into plants with blue flowers and the remainder grow into plants with white flowers. Bulbs are sold in mixed bags of 40 Russell selects a random sample of 5 bags of bulbs.

1. Find the probability that fewer than 2 of these bags will contain more bulbs that grow into plants with blue flowers than grow into plants with white flowers.
   (4) Maggie takes a random sample of \(n\) bulbs.
   Using a normal approximation, the probability that more than 244 of these \(n\) bulbs will grow into blue flowers is 0.0521 to 4 decimal places.
2. Find the value of \(n\).
   (6)
   (Total 10 marks)

10 A box contains a large number, \(n\), of identical dice, which are thought to be biased. The probability that one of these dice will show a six in a single roll is \(p\). The \(n\) dice are rolled many times and the number of sixes obtained in each trial is recorded. In \(4.01 \%\) of these trials 56 or more dice showed a six. In \(10.56 \%\) of these trials 37 or fewer dice showed a six. Using a suitable normal approximation, find the values of \(n\) and \(p\).

5 Each year a college has a large fixed number, \(n\), of places to fill. The probability, \(p\), that a randomly chosen student comes from abroad is constant. Using a suitable normal approximation and applying a continuity correction, it is calculated that the probability of more than 60 students coming from abroad is 0.0187 and the probability of fewer than 40 students coming from abroad is 0.0783 . Find the values of \(n\) and \(p\).

5 The random variable \(R\) has the distribution \(\mathrm { B } ( n , p )\).

1. State two conditions that \(n\) and \(p\) must satisfy if the distribution of \(R\) can be well approximated by a normal distribution. Assume now that these conditions hold. Using the normal approximation, it is given that \(\mathrm { P } ( R < 25 ) = 0.8282\) and \(\mathrm { P } ( R \geqslant 28 ) = 0.0393\), correct to 4 decimal places.
2. Find the mean and standard deviation of the approximating normal distribution.
3. Hence find the value of \(p\) and the value of \(n\).