Okay, so in the previous lecture, we talked about the use of confidence intervals to make descriptive inference for population parameters based on single variables. Now, we're going to present some examples of using hypothesis testing to make inferences. Again, based on descriptive quantities computed using single variables. So our first example, we're going to revisit this idea of calculating one population proportion and making inference about that proportion. We're going to look at a one-sample test for our proportion. So our research question is stated as follows, did 33% or one-third of non-Hispanic African-Americans aged 18 and above in the US in 2015-2016. Have systolic blood pressure greater than 130 mmHg, or was the population proportion different than one-third? So this is our kind of hypothesized value, that population proportion 0.33 or 33% of the target population meeting this threshold, having systolic blood pressure greater than 130 mmHg. So our inference approach, unlike with the confidence intervals, we're going to perform a one-sample test, two-tailed to either reject or fail to reject the null hypothesis that the population proportion is 0.33. So we're going to do a one-sample hypothesis test in a two-tailed testing approach where the alternative hypothesis is that the population proportion is different from 0.33. So step one in the hypothesis testing approach, we have to clearly define the null hypothesis and the alternative hypothesis, who are null is that the population proportion which were called P is 0.33. The alternative is that the population Pproportion is not equal to 0.33. So it could be larger, it could be smaller. The alternative allows the proportion to be either greater or less than 0.33. And this is again will implies performing a two-tailed test. We allow for evidence in either direction, away from that null hypothesis value. It also means that we need more evidence against the null hypothesis in order to reject it. Okay, for this particular test, we also need to set our significance level. So we're going to set our significance level at 5%, or 0.05 as a type 1 error rate. So the probability of rejecting the null hypothesis if it's actually true is 0.05. We're going to set that pretty low. And that's what we mean by the significance level. Okay so step two. Once we've clearly defined our null hypothesis all our alternative hypothesis and the significance level now, we want to compute the test statistics that's going to allow us to make a decision about the null hypothesis. So our best point estimate, assuming a simple random sample of black adults, the sample proportion is 465 divided by 1135 or 0.4097, like we saw in the previous lecture. The test statistic that we would form for testing our new hypothesis in this case, we would do this assuming that the sampling distribution of the estimated proportion is normal. We form the Z statistic as you see here. That Z statistic is defined as the point estimate 0.4097 minus the null hypothesis value which is 0.33 divided by the standard error under the null hypothesis. So notice we're using the 0.33 to estimate the standard error and the denominator in this test statistic. We use the standard error assuming the the null hypothesis halts. We want to see how unusual that best estimate is relative to the null value. Allowing for the sampling variability that we would expect to see under the null hypothesis. The resulting value of the test statistic is 5.71. Okay, so, what does that 5.71 mean? It means that the estimate is more than 5 standard errors from the 0.33 null value. That's a lot. So, that's a lot of allowing for sampling variability, our actual point estimate based on the sample is pretty far away, statistically, from the null hypothesis value. So step 3, we have our test statistic. Now we want to determine the p-value that can be used to make a decision about that null hypothesis. So if the null hypothesis were true, would a test statistic value of z = 5.71 be considered unusual enough to reject the null? Do we have enough evidence against the null given this test statistic? The P-value is the probability of seeing a test statistic of 5.71, or more extreme than that, assuming that the null hypothesis holds, or that the null hypothesis is true. Okay, so if that null hypothesis is true that test statistic z follows what's called a standard normal distribution with mean zero and variance one. You see a picture of that often, the right-hand side of this particular slide. So that test statistic follows this distribution. And we want to calculate a two-tailed test given that the alternative hypothesis is that the proportion could be greater than 0.33 or less than 0.33. So we want to see how extreme that test statistic is in both tales of this distribution. So we look at the probability of this standard normal z value being great than 5.71. Or less than negative 5.71. So, both tails. That's the whole idea of a two-tailed test because, again, we're allowing that proportion to either be greater than the null or less than the null. And the probability of being greater than 5.71 or less than negative 5.71, those areas in the far right and the far left of the tails of this distribution, that probability is essentially zero. So if the null hypothesis is true, the probability of seeing a Z statistic this large or larger is essentially zero. In other words, there is really no evidence at all in favor of the null hypothesis. It's very unusual to see a value like this if the null hypothesis 0.33 was actually true. So then, step four, we make a decision about the null. If the population proportion really was 0.33, then observing a sample proportion of 0.4097 or more extreme is highly unlikely. Since our P-value is much lower than our 0.05 significance level, remember, we chose that significance level 0.05. If our P-value drops below that stated significance level, we have evidence against the null hypothesis. And we would therefore reject the null hypothesis in this case. So if the P-value that we compute is less than the stated significance level, we have evidence against the null hypothesis. And we would reject it. So based on the estimated proportion of 0.4097, we support that the population proportion is not 0.33, and likely, the population proportion with systolic blood pressure greater than 130 mmHg is likely larger, much larger, than this 0.33. Okay, so now we're going to turn to the idea of a mean and apply the same basic approach using the hypothesis testing idea. So now our research question is was the mean systolic blood pressure for non-Hispanic African-Americans age 18 and above in the US in 2015-2016 equal to 128 mmHg or was the population mean different from 128? So, again this two tailed idea. We're allowing for the mean to be greater than 128 or possibly lower than 128. Our inference approach we're going to perform one test sample test again two tailed to either reject or fail to reject null hypothesis to the population means is 128 mm Hg for this target population. So step one again we define null and the alternative. The null is that the population mean, which we'll call mu in this case, is 128 mmHg. The alternative is that the population mean is not equal to 128 mmHg. So it could be larger, it could be smaller. Okay, so again, the alternative hypothesis allows the mean to be greater than or less than 128 mmHg. Again, we would employ a two-tail test where we need more evidence against the null hypothesis in order to reject it. Again, we'll choose a significance level of 5%, but as someone designing a study, you could choose that significance level to be whatever you want it to be, you just have to be very clear about what it is upfront. In these case again, we're choosing 5% or 0.05 type 1 aerate. Step 2, we compute the test statistic in the same general approach. Our best point estimate recall from the previous lecture, assuming a simple random sample of black adults, is that the sample mean is 128.252, so that's our best point estimate of the mean. Our test statistic then, assuming a sampling distribution of estimated means that it's normal, we use this T statistic or students' T distribution. So the best estimate is 128.252. We subtract the null value, which is 128 and then we divide by the estimated standard error. Recall that the standard deviation of blood pressure values was 19.958, and we divide that by the square root of the sample size or 1135 To get the estimated standard error. The resulting value of the t statistic is 0.425. So what does this mean? It means that our best point estimate is less than half of a standard error from the null value of 128. So we're pretty close to the null value. Allowing for sampling variability, we have more evidence in support of that null value. Step 3 given that test statistic we determine the P-value. So if the null hypothesis was true with the test statistic value of only t equal to 0.425 be unusual enough to reject the null. The P-value again is the probability of seeing a test statistic of 0.425 or more extreme assuming that the null hypotheses is true. So if the null hypothesis was true, this t statistic follows what's called a student t distribution with degrees of freedom equal to the sample size minus one or 1134. And again we use a two tailed test. Now, look at this picture. On the right hand side of the slide, student's t distribution looks a little bit like a normal distribution. They're similar in appearance, but they're also defined by degrees of freedom according to the sample size. And again for a two tailed test, we want to calculate the probability that a student t statistic with 1134 degrees of freedom is either greater than 0.425 or less than negative 0.425. So we look at the probability of being more extreme in both tails of the student t distribution. Again, given the alternative hypothesis that the mean is not 128. The result in P-value if we add up those areas in the tails of the t distribution above 0.425 or below negative 0.425, that's 0.3175. So, what that says is if the null hypothesis is true, the probability of seeing this test statistic or something more extreme is about 0.32, s pretty common that we would see something like this if the null hypothesis was actually true. So, step four, we make a decision about the null. If the population mean really was 128 mmHg, then observing a sample mean of 128.252 or more extreme is actually quite likely, unlike the case of the proportion. So since our P-value is much bigger than the 0.05 significance level. We have very weak evidence against the null hypothesis, and we would choose to fail to reject the null. Notice that we're not accepting the alternative. We're just making the decision that we fail to reject the null hypothesis, so we have evidence in support of the null hypothesis, given the small t-statistic and the large P-value. Okay, so based on our estimated mean of 128.252, we cannot support the idea that the population mean differs from 128 mmHg. We have support in favor of the null hypothesis that that mean is certainly a plausible value for the population mean. So what's next? We've talked about a confidence interval approach for descriptive parameters. And we've talked about a hypothesis testing approach for descriptive parameters. Next, we're going to talk about how to make inferences about differences between subgroups, so not for overall population means or proportions, but rather differences between subgroups. So we'll talk about forming confidence intervals for differences in means and proportions and we'll talk about hypothesis testing approaches for comparing means and proportions.
