Justify normal approximation

5 In the holidays Martin spends \(25 \%\) of the day playing computer games. Martin's friend phones him once a day at a randomly chosen time.

1. Find the probability that, in one holiday period of 8 days, there are exactly 2 days on which Martin is playing computer games when his friend phones.
2. Another holiday period lasts for 12 days. State with a reason whether it is appropriate to use a normal approximation to find the probability that there are fewer than 7 days on which Martin is playing computer games when his friend phones.
3. Find the probability that there are at least 13 days of a 40-day holiday period on which Martin is playing computer games when his friend phones.

2 In Scotland, in November, on average \(80 \%\) of days are cloudy. Assume that the weather on any one day is independent of the weather on other days.

1. Use a normal approximation to find the probability of there being fewer than 25 cloudy days in Scotland in November (30 days).
2. Give a reason why the use of a normal approximation is justified.

2 There is a probability of \(\frac { 1 } { 7 }\) that Wenjie goes out with her friends on any particular day. 252 days are chosen at random.

1. Use a normal approximation to find the probability that the number of days on which Wenjie goes out with her friends is less than than 30 or more than 44.
2. Give a reason why the use of a normal approximation is justified.

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.

8 A sales office employs 21 representatives. Each day, for each representative, the probability that he or she achieves a sale is 0.7 , independently of other representatives. The total number of representatives who achieve a sale on any one day is denoted by \(K\).

1. Using a suitable approximation (which should be justified), find \(\mathrm { P } ( K \geqslant 16 )\).
2. Using a suitable approximation (which should be justified), find the probability that the mean of 36 observations of \(K\) is less than or equal to 14.0 . 4

3. The discrete random variable \(X\) is distributed \(\mathrm { B } ( n , p )\).

1. Write down the value of \(p\) that will give the most accurate estimate when approximating the binomial distribution by a normal distribution.
2. Give a reason to support your value.
3. Given that \(n = 200\) and \(p = 0.48\), find \(\mathrm { P } ( 90 \leq X < 105 )\).

15 A fair dice is thrown 1000 times and the number, \(X\), of throws on which the score is 6 is noted.

1. 1. State the distribution of \(X\).
   2. Explain why a normal distribution would be an appropriate approximation to the distribution of \(X\).
2. Use a normal distribution to find two positive integer values, \(a\) and \(b\), such that \(\mathrm { P } ( a < X < b ) \approx 0.4\).
3. For your two values of \(a\) and \(b\), use the distribution of part (a)(i) to find the value of \(\mathrm { P } ( a < X < b )\), correct to 3 significant figures. \section*{OCR} Oxford Cambridge and RSA