For part A I have tried 0.7324 and 0.7325 and both are wrong. In the following p

Question

For part A I have tried 0.7324 and 0.7325 and both are wrong.

In the following problem, check that it is appropriate to use the normal approximation to the binomial. Then use the normal distribution to estimate the requested probabilities It is known that 75% of all new products introduced in grocery stores fail (are taken off the market) within 2 years. If a grocery store chain introduces 65 new products, find the following probabilities. (Round your answers to four decimal places.) (a) within 2 years 47 or more fail 0.7324 x (b) within 2 years 58 or fewer fail 0.9974 (c) within 2 years 15 or more succeed 0.6915 (d) within 2 years fewer than 10 succeed 0.0268

Explanation / Answer

X ~ Binomial (n,p)

Where n = 65, p = 0.75

Mean = n * p = 0.75 * 65 = 48.75

Standard deviation = Sqrt( n p ( 1 - p) )

= Sqrt( 65 * 0.75 * 0.25)

= 3.4911

a)

Using normal approximation to binomial distribution with continuity correction,

P(X >= x) = P( X > x - 0.5) (Using continuity correction ) and

P( X >= x) = P( X > x - 0.5) = P( Z > x - 0.5 - Mean / Standard deviation) (Using normal approximation)

Therefore,

P( X >= 47) = P( X > 46.5)

= P( Z > 46.5 - 48.75 / 3.4911)

= P( Z > -0.6445)

= P( Z < 0.6445)

= 0.7404

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