Question

As in the previous problem, a fair coin is flipped 28 times. If x is the number of heads, then the distribution of x can be approximated with a normal distribution, n(14,2.6), where the mean (μ) is 14 and standard deviation (σ) is 2.6. What is the probability that the number of heads is less than or equal to 10?

1) 0.001
2) 0.022
3) 0.115
4) 0.841

Answer 1

The probability of getting 10 or fewer heads when flipping a fair coin 28 times, approximated by a normal distribution with a mean of 14 and a standard deviation of 2.6, corresponds to a z-score of -1.54. The closest probability found from the standard normal distribution is approximately 2.2%.

Step-by-step explanation:

Using the normal distribution approximation to the binomial, with a mean (μ) of 14 and a standard deviation (σ) of 2.6, we want to find the probability that the number of heads is less than or equal to 10. To do this, we calculate the z-score which standardizes the value of 10 using the given mean and standard deviation:

z = (X - μ) / σ

For X = 10,
z = (10 - 14) / 2.6 = -4 / 2.6 = -1.54

After calculating the z-score, we look it up in the standard normal distribution table, or use a calculator that provides the cumulative probability for a standard normal distribution up to a z-score of -1.54. This gives us the probability of getting 10 or fewer heads. The options provided do not precisely match standard z-table values, but we can approximate and find the closest probability, which is option 2) 0.022.

Thus, the probability of getting less than or equal to 10 heads is approximately 2.2%.