Answer :
The purpose of checking the Large Counts condition when performing a one-sample z test for p is:
Option c) To make sure that the sample size is large enough for the sampling distribution of p-hat to be approximately Normal.
What is the purpose of checking the Large Counts condition when performing a one-sample z test for p?
This is because the Large Counts condition is used to determine if the sample size is large enough for the Central Limit Theorem to apply, which states that the sampling distribution of p-hat will be approximately Normal if the sample size is large enough. In order to check this condition, we need to make sure that both np and n(1-p) are greater than or equal to 10. If this condition is met, then we can assume that the sampling distribution of p-hat is approximately Normal and we can proceed with the one-sample z test for p.
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The purpose of checking the Large Counts condition when performing a one-sample z test for p is (c) To make sure that the sampling distribution of p-hat is approximately Normal.
The z test relies on the assumption that the sampling distribution of the sample proportion is Normal to calculate the z-score and make inferences about the population proportion. In detail:
- A one-sample z test for a population proportion (p) involves taking a simple random sample from the population where each trial has two outcomes: success or failure, with the same probability of success (p).
- To use the z test, we need to ensure that the shape of the binomial distribution (which models the number of successes in the sample) is similar to the shape of the Normal distribution. This makes it possible to use the properties of the Normal distribution for hypothesis testing.
- The Large Counts condition requires that both np and n(1-p) (where n is the sample size and q = 1 - p) are greater than 5. That is:
[tex][ np > 5 \ n(1 - p) > 5 \[/tex] - Meeting these conditions ensures that the binomial distribution of the sample proportion can be well approximated by the Normal distribution, allowing us to use z-scores for hypothesis testing.
