need help with this Math question!

Question

In the 2016 referendum on the independence of wales from the UK, approx.

44% of welsh population was pro-independence. Suppose we now interview a

random sample of 100 people. What is the probability that 50 or more of them

are pro-independence? (assuming that the welsh population preference didn’t

change since 2014)

think i should apply the normal approximation of binomial but i don't know exactly how to or what the right answer i'm stuck

Answer 1

Hi Mo,

You are on the right track so far. To use the normal approximation of the binomial, you have to meet 2 conditions (n is the sample size, which is 100 in this scenario, and p is the desired probability, or 0.44 for pro-Independence):

1. np ≥ 10 (this condition is met because 100 * 0.44 = 44, which is greater than 10)
2. n(1 - p) ≥ 10 (this condition is also met because 100 * (1 - 0.44) = 56)

Remember that normal distribution uses the mean (μ) and standard deviation (σ) while binomial distribution uses n and p. To convert from binomial to normal, we use the following calculations

μ = np = 44 (this is the same as the number of people that we expect to vote pro-independence).

σ = √(n * p * (1 - p)) = 4.96

On your calculator, go to the "Normal Distribution CDF" option (it might be called something different on yours), and substitute the above values for mean and standard deviation. Use a minimum bound of 50 and a maximum bound of 100, as we want to find the probability that at least 50 people are pro-independence. Once you calculate this, you should get a probability of 0.113 that at least 50 people are for Welsh Independence.

I hope this helps! Feel free to ask any questions.

Akshat Y.

Answer 2

Normal approximation has mean = np, standard deviation = √ npq

Here n = 100, p = 0.44 q = 1-p = 0.56

so mean is 44, sd is 4.96 approx

So to find pr (X>50). On a TI-84 NORMALCDF. lower = 50, upper = 1E99, mean = 44, sd = 4.96

-----------------------------------------------------------------------------------------------------------------------------------------