Question

A fair coin is flipped 40 times. Let x be the number of heads. Which normal distribution best approximates x? Round to one decimal place if entering a decimal answer below.

1) Binomial Distribution
2) Poisson Distribution
3) Uniform Distribution
4) Normal Distribution

Answer 1

The normal distribution approximates the number of heads when flipping a fair coin 40 times, with the binomial distribution originally describing it. This approximation is valid when both np and nq are greater than or equal to 5, resulting in a mean of 20 and a standard deviation of approximately 3.2.

Step-by-step explanation:

The normal distribution best approximates x, the number of heads from flipping a fair coin 40 times. In this scenario, we use the binomial distribution because the flips are independent, and each has only two outcomes: heads or tails, with equal probability (p=q=0.5). However, when the number of trials (n) is large, and both np and nq are greater than or equal to 5, the binomial distribution can be approximated by a normal distribution.

To calculate the mean (μ) and standard deviation (σ) of the normal distribution, use μ = np and σ = √npq. In this case, the mean would be μ = 40(0.5) = 20 and the standard deviation σ = √(40)(0.5)(0.5) = √10. Therefore, the normal distribution that approximates the number of heads would have a mean of 20 and a standard deviation of approximately 3.2 (rounded to one decimal place).