Question

If =4/5, what is the smallest value of n that satisfies the requirements for a normally distributed p^ (p-hat)?

Answer 1

The smallest value of n that satisfies the requirements for a normally distributed p-hat, given p = 4/5, is 16.

Step-by-step explanation:

In statistical terms, when dealing with proportions, the distribution of the sample proportion (p-hat) becomes approximately normal when the sample size is sufficiently large. The criteria for this approximation often involve two conditions: the sample size must be large enough, and the conditions for a binomial distribution must be met.

For a binomial distribution, the conditions are generally stated as np ≥ 10 and n(1-p) ≥ 10, where n is the sample size and p is the probability of success. In this case, p is given as 4/5. To find the smallest value of n, we need to satisfy both conditions.

Using np ≥ 10:

`\[ n * (4)/(5) \geq 10 \]`

`\[ n \geq (50)/(4) \]`

`\[ n \geq 12.5 \]`

Using n(1-p) ≥ 10:

`\[ n * (1)/(5) \geq 10 \]`

`\[ n \geq 50 \]`

To satisfy both conditions, the minimum value for n is 16, ensuring that the sample size is large enough for the normal approximation to the sampling distribution of the sample proportion to be valid.Answer: