(a) Suppose n = 26 and p = 0.30. Can we approximate p̂ by a normal distribution? Why? (Use 2 decimal places.)
To determine if we can approximate p̂ by a normal distribution, we need to check if the conditions for approximation are met.
The conditions for approximating p̂ by a normal distribution are as follows:
1. np ≥ 10
2. n(1 - p) ≥ 10
Let's plug in the values n = 26 and p = 0.30 into these conditions:
1. np = 26 * 0.30 = 7.80
2. n(1 - p) = 26 * (1 - 0.30) = 18.20
Based on these calculations, we can see that both conditions are not met. In particular, np (7.80) and n(1 - p) (18.20) are both below 10.
Therefore, we cannot approximate p̂ by a normal distribution when n = 26 and p = 0.30 because the conditions for approximation are not satisfied.