1	Inference about a population proportion BPS chapter 20
2	Objectives (BPS chapter 20) Inference for a population proportion The sample proportion The sampling distribution of Large sample confidence interval for p Accurate confidence intervals for p Choosing the sample size Significance tests for a proportion
3	The two types of data — reminder Quantitative Something that can be counted or measured and then added, subtracted, averaged, etc., across individuals in the population. Example: How tall you are, your age, your blood cholesterol level Categorical Something that falls into one of several categories. What can be counted is the proportion of individuals in each category. Example: Your blood type (A, B, AB, O), your hair color, your family health history for genetic diseases, whether you will develop lung cancer
4	The sample proportion We now study categorical data and draw inference on the proportion, or percentage, of the population with a specific characteristic. If we call a given categorical characteristic in the population “success,” then the sample proportion of successes, ,is:
5	Sampling distribution of The sampling distribution of is never exactly normal. But as the sample size increases, the sampling distribution of becomes approximately normal.
6	Implication for estimating proportions The mean and standard deviation (width) of the sampling distribution are both completely determined by p and n. Thus, we have only one population parameter to estimate, p.
7	Conditions for inference on p Assumptions: We regard our data as a simple random sample (SRS) from the population. That is, as usual, the most important condition. The sample size n is large enough that the sampling distribution is indeed normal. How large a sample size is enough? Different inference procedures require different answers (we’ll see what to do practically).
8	Large-sample confidence interval for p Use this method when the number of successes and the number of failures are both at least 15.
9	Medication side effects
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12	“Plus four” confidence interval for p A simple adjustment produces more accurate confidence intervals. We act as if we had four additional observations, two being successes and two being failures. Thus, the new sample size is n + 4 and the count of successes is X + 2.
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14	Choosing the sample size You may need to choose a sample size large enough to achieve a specified margin of error. However, because the sampling distribution of is a function of the population proportion p this process requires that you guess a likely value for p: p*.
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16	Significance test for p The sampling distribution for is approximately normal for large sample sizes, and its shape depends solely on p and n. Thus, we can easily test the null hypothesis: H₀: p = p₀ (a given value we are testing)
17	P-values and one- or two-sided hypotheses — reminder
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20	Interpretation: magnitude versus reliability of effects The reliability of an interpretation is related to the strength of the evidence. The smaller the P-value, the stronger the evidence against the null hypothesis and the more confident you can be about your interpretation. The magnitude or size of an effect relates to the real-life relevance of the phenomenon uncovered. The P-value does NOT assess the relevance of the effect, nor its magnitude. A confidence interval will assess the magnitude of the effect. However, magnitude is not necessarily equivalent to how theoretically or practically relevant an effect is.