Inference for Single Proportions and Means T.Scofield
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1 Inference for Single Proportions and Means TScofield Confidence Intervals for Single Proportions and Means A CI gives upper and lower bounds between which we hope to capture the (fixed) population parameter we are studying (perhaps a population proportion p, a population mean µ, a difference of proportions or means p 1 p 2, µ 1 µ 2, a population correlation ρ, etc) Our sample gives us a point estimate (ˆp, x, p 1 p 2, x 1 x 2, r, etc) For all instances besides the percentile confidence interval construction described in Section34, our confidence interval is (point estimate) ± (margin of error) Our text makes several passes at how to calculate the margin of error, each time refining what was done earlier Section 33 In this first pass at confidence interval construction, we only discussed the construction of 95% CIs Without having a direct way to simulate the sampling distribution of our point estimate (ie, sample statistic), we construct bootstrap distributions which, though centered at the sample statistic instead of the population parameter, have roughly the same shape and spread as the sampling distribution We ask that the bootstrap distribution appear symmetric, bell-shaped If it is, we proceed, noting the standard error SE (ie, the standard deviation of the sampling distribution) is approximately the same as the standard deviation of the bootstrap distribution We compute the latter, and then take (margin of error) = (20) (standard deviation of the bootstrap distribution) Section 52 Following Section 51, we are ready for a second pass at confidence interval construction The Central Limit Theorem tells us that we can expect the sampling distributions for ˆp and x will be symmetric, bell-shaped normal, in fact so long as our sample size is large enough This normality should be reflected in bootstrap distributions, and we acquired a new tools, rules of thumb (np 10 and n(1 p) 10 for proportions, or n 30 for means), and the quantile-quantile plot (produced via the qqmath() command), to help us decide if the sample size is large enough We also learn that one need only go 196 standard deviations, away from the mean (not a full two of them, as in Section 33) to enclose 95% of observations in a normal distribution, a fact revealed by the command qnorm(0975) ## [1] Thus, we now construct 95% CIs using (margin of error) = (196) (standard deviation of the bootstrap distribution) The number 196 is called the critical z-value for 95% CIs, and is often labeled as z Similar sorts of qnorm() commands may be used to find the critical value appropriate for other levels of confidence For example, we use qnorm(095) 1
2 ## [1] to obtain z for a 90% CI Having determined the appropriate choice of z for the particular level of confidence desired, our confidence interval is (point estimate) ± z (standard deviation of the bootstrap distribution) Note that we still need to produce a bootstrap distribution in this process Section 62 (modifications to CIs for p only) This section focuses exclusively on a refinement to the procedure for constructing confidence intervals for a population proportion p We learned, in Section 61, a formula for standard error p(1 p) SEˆp = n We are reminded, of course, that our CIs are premised on the sampling distribution for ˆp being normal, which it is reasonable to assume to be the case if the rules of thumb np 10, n(1 p) 10, are met There is the problem that we do not actually know p, so instead we check that nˆp 10, n(1 ˆp) 10 If these check out, we take the margin of error to be ˆp(1 ˆp) (margin of error) = z, n choosing z as we learned to do in Section 52 Note that we do not need to produce a bootstrap distribution in this process, as we obtain the standard error from a formula instead Section 65 (modifications to CIs for µ only) This section focuses exclusively on a refinement to the procedure for constructing confidence intervals for a population mean µ In Section 64, we learn this formula, used to approximate the standard error SE x = s n, where s is the sample standard deviation The actual standard error, σ/ n, would have allowed us to use a critical z-value as before Since, we must replace the stability of a fixed population standard deviation σ with the variability of the sample standard deviation s, we turn to a new type of critical value, called t, obtained from an appropriate t-distribution Once again, we want assurances that the sampling distribution for x, our point estimate, is normal which, according to the text, is reasonable to assume true if the sample size n 30 For a sample of size n = 40 that yields mean and standard deviation x and s respectively, we obtain t using the qt() command For a 95% CI, we would take t to be the value qt(0975, df=39) ## [1]
3 so that the 95% CI is x ± (2022) s 40 If, instead, the sample size is n = 34 and we want a 99% CI, we take t to be qt(0995, df=33) ## [1] Whatever the specific details leading to t, our CI is built via the formula x ± (t ) s n As in Section 62 (for proportions), we do not need to produce a bootstrap distribution, as we obtain the standard error from a formula instead 3
4 Hypothesis Testing for Single Proportions and Means From the start we have outlined the steps of hypothesis testing as these: 1 Identify the research question, along with relevant variables 2 Formulate hypotheses (null and alternative) appropriate to the research question 3 Obtain a random sample, and a corresponding sample statistic (called the test statistic) 4 Determine the null distribution ie, the sampling distribution of the test statistic assuming the null hypothesis is true 5 Determine the P -value ie, the likelihood that your test statistic, or something even more extreme, occurs under the null hypothesis 6 Draw a conclusion (ie reject H 0 or not) Of these, only Steps 4 and 5 undergo modifications as we make successive passes to refine the process Here is a breakdown Section 44 In this first pass, we generate a randomization distribution, which simulates the null distribution At this time we advised a check for normality, but had not yet learned the rules of thumb for proportions (np 10 and n(1 p) 10) or for means (n 30) which are often considered sufficient to ensure normality; in hindsight, a quantile-quantile plot using qqmath() may be advisable, as well, particularly when we are doing hypothesis testing on a mean Along with normality, we should make sure the randomization distribution is centered at the value proposed in the null hypothesis which, in the case of hypothesis testing on a population mean, generally involves shifting a bootstrap distribution to the correct location To get the P -value, we locate the test statistic in the randomization distribution, then calculate the proportion of randomization statistics in one or both tails as the alternative hypothesis dictates This should always be a number between 0 and 1 Section 52 Here, our modification is to Step 5 only Having generated a randomization distribution (our stand-in for the null distribution of Step 4) and observed it to be normal, we calculate its standard deviation, which approximates the standard error (SEˆp or SE x ) Then, in the case our alternative hypothesis is one-sided and left-tailed (perhaps H a : µ < µ 0, if we are testing a mean), our approximate P -value comes from the command pnorm( x, mean = µ 0, sd = SE x ) If our alternative hypothesis is one-sided and right-tailed (H a : p > p 0, for a proportion), then 1 - pnorm(ˆp, mean = p 0, sd = SEˆp ) generates the approximate P -value These commands can be used, with appropriate alterations, to yield the approximate P -value when hypothesis is 2-sided (H a : µ µ 0, for instance) Again, the resulting P -value should be a number between 0 and 1 Section 63 Let us assume the null hypothesis is H 0 : p = p 0 Using the hypothesized value p 0 for the population proportion, if the rules of thumb np 0 10 and n(1 p 0 ) 10 are met, then we take the null distribution in Step 4 to be Norm(p 0, p 0 (1 p 0 )/n) As in Section 62, this means we do not need to generate a randomization distribution to find the standard error; it comes from the formula We can find (directly) the 4
5 appropriate P -value using pnorm() commands, as in Section 52 For instance, if the alternative hypothesis were one-sided and left-tailed (ie, H 0 : p < p 0 ), then the P -value would come from the command pnorm( ˆp, p 0, p 0 (1 p 0 )/n) where, of course, it is necessary to fill in the correct values Of course, since ˆp is from a normal distribution, one can convert it to a Z-score ˆp p 0 z = p0 (1 p 0 )/n, and then you can use the simpler command pnorm(z) using this computed z Section 66 As we now have a formula to estimate SE x, we are free of needing to produce a randomization distribution, at least if we are reasonably sure that x has a normal sampling distribution (n 30?) We assume, as before, that the null hypothesis is H 0 : µ = µ 0 We must standardize the test statistic, and use a t-distribution calculator to find the P -value The standardized value we call t (not z) t = x µ 0 s/ n If the alternative hypothesis is one-sided, left-tailed, then the P -value is pt(t, df=n-1) where t is the standardized value indicated above, and n is the sample size 5
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