Inference for Population Proportion

Transcription

1 Inference for Population Proportion The objective is to make inference about (unknown) population proportion p, based on sample proportion ˆp from the data. For example, we may wish to know the proportion (or percentage) p of all potential voters who will support candidate A in an election or support a new policy. The population here can be viewed as binary (discrete): each individual has only two possible outcomes, denoted by X i = 1 or X i = 0 (such as Yes/No, happy/unhappy, pass/fail ).

2 Inference for Population Proportion We can make inference about population proportion p based on the sample proportion ˆp: ˆp = X n, where X is the total number of cases with X i = 1 and n is the total sample size. Note that X B(n,p), but the distribution of ˆp is unknown for small sample sizes (for large sample sizes, the distribution of ˆp is approximately normal). We can make statistical inference about p in two ways: construct a confidence interval for p, perform a hypothesis testing for p.

3 Inference for Population Proportion Inference for population proportion p is based on the sampling distribution of the sample proportion ˆp: When np 10 and n(1 p) 10 (i.e., n is large and p is not too close to 0 or 1), we have, approximately, ˆp N ( p, ) p(1 p), or n ˆp p p(1 p) n N(0,1). Note that E(ˆp) = p and SD(ˆp) = p(1 p) n. The above approximate normal distribution is then used to construct confidence intervals or hypothesis testing for p.

4 CI for Population Proportion An approximate (1 α) 100% confidence interval (CI) for the population proportion p is ( ) ˆp z ˆp(1 ˆp) α, ˆp + z ˆp(1 ˆp) α, or ˆp ± z α SE(ˆp), 2 n 2 n 2 where ˆp is the sample proportion and z α/2 is the 1 α 2 percentile of N(0,1). For example, for a 95% CI, α = 0.05 and z α/2 = z = The term z ˆp(1 ˆp) α 2 n (or z α SE(ˆp)) is called margin of error. 2

5 CI for Population Proportion Interpretation of a 95% CI for p: if we take many samples of size n from the same population and construct a CI for p based on each sample using the same formula, about 95% of all these CI s will cover (include) the true value p. Similar interpretation for a 99% CI. We call 95% (or 99%) the confidence level. For a given confidence level, the shorter the CI the more accurate the interval estimation for p.

6 CI for Population Proportion The length (or width) l of the CI for p is l = 2z α 2 ˆp(1 ˆp) n = 2 marginal of error. Thus, the length of the CI depends on the sample size n and confidence level 1 α. The above formula can be used to compute a desirable sample size n (before collecting data), with a pre-specified length l and confidence level 1 α, where ˆp may be replaced by a guessed value (e.g., 0.5).

7 Inference for Population Proportion Consider one-sided hypotheses: H 0 : p = p 0 (or p p 0 ) versus H a : p > p 0, where p 0 is a known proportion. For example, H 0 : p 0.5 vs H a : p > 0.5 (here p 0 = 0.5). If H 0 holds, we have the following approximate sampling distribution for ˆp, when the sample size n is large (i.e., when np 10 and n(1 p) 10) ( ) p0 (1 p 0 ) ˆp N p 0,, or n ˆp p 0 p 0 (1 p 0 ) n N(0,1).

8 Inference for Population Proportion The test statistic is given by Z = ˆp p 0 p 0 (1 p 0 ) n Under H 0, we have the null distribution Z N(0,1) approximately for large samples. At significance level α, we reject H 0 : p = p 0 (or p p 0 ) if the value of the test statistic z > z α. So the where z α is the 1 α percentile of N(0,1). So the rejection region is all z values satisfying {z > z α}..

9 Inference for Population Proportion Alternatively, the p-value is given by p value = P(Z z), where Z N(0,1) is a random variable and z is the value of the test statistic computed from a data/sample. The p-value is the probability of observing the value z (computed from a dataset) or observing more extreme values (i.e., values larger than z here) if the null hypothesis holds. The smaller the p-value, the stronger the evidence against H 0 (so in favor of H a ).

10 Inference for Population Proportion There are another form of one-sided hypotheses H 0 : p p 0 versus H a : p < p 0 and two-sided hypotheses H 0 : p = p 0 versus H a : p p 0, where p 0 is a known proportion. The choice of hypotheses should be determined by the research questions. For the above hypotheses, the formula of the test statistic remains the same. However, the formulas for the rejection regions and p-values need to be modified, depending on the form of H a (please see lecture notes for details).

11 General comments about hypothesis testing General comments about hypothesis testing: It is important to set up the appropriate hypotheses, i.e., one sided or two sided hypotheses. The types of hypotheses are determined by the research questions, and should be set up before you see the data. Usually the alternative hypothesis H a is what a researcher tries to verify (or something new), while H 0 is the complement of H a.

12 General comments about hypothesis testing The hypotheses are formulated in terms of population parameters (e.g., population proportion p), not statistics. The test statistic may be viewed as a standardized difference between the sample proportion ˆp and true population proportion p if the null hypothesis H 0 holds. The decision for a hypothesis testing problem can be (i) rejecting H 0 or not based on a given significance level α; or (ii) any evidence against H 0 or not based on a p-value; or reporting both results. Interpret the decision in words in the context of a given application.

13 Relationship between two-sided test and CI A 95% CI covers p 0 if and only if the two-sided test for H 0 : p = p 0 vs H a : p p 0 fails to reject H 0 at level 5% level. Or, equivalently, the two-sided test for H 0 : p = p 0 vs H a : p p 0 rejects H 0 at level 5% level if and only if the 95% CI does not cover p 0. Notice the correspondence between 95% and 5%. Similar statements hold for 99% or 90% CIs, corresponding to 1% or 10% significance levels. Examples: if a 95% (or 99%) CI for p is (0.1, 0.4), then a test for H 0 : p = 0.5 vs H a : p 0.5 will reject H 0 at level 5% (or 1%), or vice versa.

14 Relationship between significance level α and p-value Relationship between p-value and significance level α: A test rejects H 0 at level α if and only if its p-value is less than α. Example: if a p-value=0.03, then we reject H 0 at 5% level (but not at 1% level). Or, if we reject H 0 at 5% level, the p-value must be smaller than Thus, a p-value is more informative than a rejection region, since a rejection region is determined by a specific significance level.

15 Hypothesis Testing Since a sample is only a small subset of the population, statistical inference cannot lead to definite (100%) conclusions. That is, errors are inevitable. In hypothesis testing, there are two types of errors: type I and type II errors. Neither errors will be 0. Type I error: we reject H 0 when H 0 is in fact true. Type II error: we accept H 0 when H 0 is in fact false (i.e. H a is true).

16 Hypothesis Testing The power of a test is the probability of rejecting H 0 when an alternative is true, i.e., the ability to detect an alternative hypothesis we wish to prove. Power = 1 P(type II error). Significance level α = P(type I error). The power depends on (i) the sample size n, (ii) the significance level α, (iii) the type of hypotheses (one-sided or two-sided), and (iv) the population standard deviation. See an example in Chapter 6.

17 Hypothesis Testing In hypothesis testing, the test statistic and its null distribution are evaluated under the null hypothesis (i.e., assuming H 0 holds). Thus, our statements are usually, reject the null hypothesis or fail to reject the null hypothesis or strong evidence against the null hypothesis, etc. Here, null hypothesis cannot be replaced by alternative hypothesis when we make a statement about the result of the test.

18 Hypothesis Testing Sample size calculation: In designing a study, before collecting data, we first need to choose the sample size n. This is usually based on the desirable power we hope to have (e.g., 80% power), or the desirable length (or margin of error) of a confidence interval. For example, if we wish to have at least 80% power to detect a specific alternative, such as detecting p = 0.6 in testing H 0 : p 0.5 versus H a : p > 0.5, how large should the sample size n be?