Question

53% of people beleive in love at first sight. Suppose we take a random sample of 200 people. What is the probability that

1 a ) Exactly 95 of them beleive in love at first sight

1 b ) 120 or more of them beleive in love at first sight

1 c) Fewer than 90 of them beleive in love at first sight

Answer 1

P( People believe in love at first sight) = 0.53 = p

Sample size, n= 200

Let X be the number of people who believe in love at first sight

X~ Binomial ( 200, 0.53)
Since the sample size is large we will use normal approximation

X~ Normal ( np, np(1-p))

X~ Normla ( 106, 49.82)

a)P( X=95) = P( 94.5 < X < 95.5) ; Using the continuity correction

= P( <   < )

= P( -1.63 < z < -1.49)

= P( z < -1.49) - P( z < -1.63)

= [ 1- P( z < 1.49) ] - [ 1- P( z < 1.63)]

= P( z < 1.63) - P( z < 1.49)

= 0.94845- 0.93189

= 0.01656

b) P( X >= 120) = P( X > 119.5) ; using continuity correction

= P( > )

= P( z >1.91)

= 1- P( z < 1.91)

= 1- 0.97193

= 0.02807

c) P( X < 90) = P( X < 89.5) ; using continuity correction

= P( < )

= P( z < -2.34)

= 1- P( z < 2.34)

= 1- 0.99036

= 0.00964