Question

Use the normal distribution to approximate the following binomial distribution. A soccer player has a 65% chance of making a free kick. What is the probability of him making fewer than 24 free kicks in 40 kicks?

a)0.2546

b)0.2119

c)0.2033

d)0.3085

e)0.1611

f) None of the above.

Answer 1

Answer:-

Given That:-

Use the normal distribution to approximate the following binomial distribution. A soccer player has a 65% chance of making a free kick. What is the probability of him making fewer than 24 free kicks in 40 kicks?

Given,

X ~ Binomial(n = 40, p = 0.65)

Binomial can be approximated to normal with

= 40 * 0.65

= 26

= 3.016621

   3.017

P(X < 24)

Since we are approximating a discrete binomial distribution by continous normal distribution, values between 23.5 and 24.5 both approximation to 24. Thus "less than 24" corresponds to continuous normal distribution with X < 23.5 after continuity correction.

Normal Distribution x = 23.5

,

We convert this to standard normal using

P(X < 23.5) = Area to the left of 23.5

,

P(X < 23.5) = P(Z < -0.83)

= 0.2033 (from z - table)

P(X is less than 24) = 0.2033

If excel is used instead of z - table, it would be: 0.2036252560

Note that if this question is solved without continuity correction, the answer would have been:

P(X < 24) = P(Z < -0.66)

= 0.2546

Without integer - continuity correction:

P(X < 24) = 0.2546

If excel is used instead of z - table, it would be: 0.2536673707

The correct answer is option (a)

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