What is p hat?

p̂ (p-hat) is a point estimator that represents the sample proportion. It's used to estimate the population proportion, denoted by p.

• Definition: p̂ is calculated as the number of successes (the events of interest) in a sample divided by the total sample size (n). p̂ = x / n where 'x' is the number of successes.

• Usage: p̂ is most often used in hypothesis testing and confidence intervals related to proportions.

• Properties:

  • p̂ is an unbiased estimator of p (meaning on average, it will estimate the true population proportion).
  • The standard error of p̂ is estimated as sqrt((p̂ * (1 - p̂)) / n). This is essential for constructing confidence intervals and performing hypothesis tests.

• Assumptions: Using p̂ effectively relies on certain assumptions:

  • The sample is a random sample from the population.
  • The sample size is large enough (typically, both np and n(1-p) are greater than or equal to 10, although some sources suggest 5). This ensures that the sampling distribution of p̂ is approximately normal, enabling the use of normal distribution based statistical inference.

• Example: If you survey 200 people and find that 120 of them prefer a certain product, then p̂ = 120 / 200 = 0.6. This is your estimate of the proportion of people in the entire population who prefer that product.