DATASCI 6.1

What is the sampling distribution of p^?

The distribution of sample proportions from many repeated samples of size n. It describes how p^ varies from sample to sample.

When is the sampling distribution of p^p^ approximately normal?

Observations are independent, Success-failure condition: np≥10np≥10 and n(1−p)≥10

What is the mean of the sampling distribution of p^?

µp^ = p

What value is used to check condition for a confidence interval?

Use p^

What value is used to check conditions for a hypothesis test?

Use p0 (null value)

What is a confidence interval?

A range of plausible values for a population parameter

What are the 4 steps for constructing a confidence interval?

1. Prepare: identify p^, n, confidence level, 2. Check: conditions using p^, 3. Calculate: SE, z star, interval, 4. Conclude: interpret in context

When should you use a confidence interval

When the question ask to estimate or give a range

What is a one-proportion z test

A hypothesis test used to evaluate claims about a population proportion

What are the 4 steps for a hypothesis test

1. Prepare: parameters, hypotheses, a, data, 2. Check: conditions using p0, 3. Calculate: z, p-value, 4. Conclude: compare p-value to a

Decision rule for hypothesis testing

If p ≤ a -> reject H0. If p > a -> Fail to reject H0

When should you use a hypothesis test?

When the question asks to test a claim or compare values

What does independent mean in this context

Observations are not related (random sample or < 10% of population)

Why is the success-failure condition important?

It ensures the sampling distribution is approximately normal, allowing use of Z methods

What do we do if success-failure condition fails in hypothesis testing?

Use simulation with the null value p0

What do we do if success-failure condition fails for confidence intervals?

Use Clopper-Pearson interval

Why is choosing a large sample size important?

It reduces the margin of error, making estimates more precise

Relationship between sample size and margin of error?

Larger n -> smaller SE -> smaller ME. Smaller n -> larger SE -> Larger ME

Key differences between CL and hypothesis test?

The confidence interval estimates a range using p^. Hypothesis test evaluates a claim using p0