Under what circumstances does the sampling distribution of the proportion approximately follow the normal distribution?
If the criterion being measured varies by chance.
The sampling distribution of the proportion approximately follows the normal distribution under the following circumstances:
1. Large sample size: The sample size should be sufficiently large. As a general guideline, when the sample size, denoted as n, is greater than or equal to 30, the sampling distribution can be approximated to a normal distribution.
2. Independent and identically distributed (i.i.d.) observations: The observations within the sample should be independent of each other and come from the same population. This assumption ensures that each observation contributes equally to the calculation of the proportion.
3. Sufficient success and failure counts: The number of successes and failures in the sample should be reasonably large. Specifically, the number of successes (n*p) and the number of failures (n*(1-p)) should both be greater than or equal to 10, where p represents the true population proportion.
Under these circumstances, the sampling distribution of the proportion can be reasonably approximated by a normal distribution using the Central Limit Theorem. This allows for the use of various statistical methods that rely on the assumption of normality.
The sampling distribution of the proportion follows an approximately normal distribution under the following circumstances:
1. Random Sampling: The sample is selected randomly from the population.
2. Binomial Distribution: The variable being measured is categorical and can take only two possible outcomes (e.g., success or failure, yes or no).
3. Independent Trials: Each observation in the sample is independent of the others. The outcome of one trial does not affect the outcomes of other trials.
4. Success-Failure Rule: The number of successes and failures in the sample should be reasonably large. Typically, at least 10 successes and 10 failures are required for the normal approximation to be valid.
When these conditions are met, the Central Limit Theorem (CLT) ensures that the sampling distribution of the proportion will be approximately normal. The Central Limit Theorem states that the distribution of the sample means will approach a normal distribution as the sample size increases, even if the population distribution is not normal.
To apply the Central Limit Theorem and check if the sampling distribution of the proportion is approximately normal, you will need to collect a random sample, determine the number of successes and failures in the sample, and ensure that the success-failure rule is met. Then, calculate the proportion of successes in the sample (p-hat). Finally, you can use statistical software or tools to plot a histogram or calculate the z-score to assess whether the distribution is approximately normal.