Like the judge in the TV show, if we have enough evidence to conclude that the null is not true, we can reject the null. Jin explained that if the judge had enough evidence to conclude the person on trial was not innocent she would have. The judge specifically stated that she did not have enough evidence to reject innocence (the null hypothesis).

When the judge acquits a defendant, as on the T.V. show, this does not mean that the judge accepts the defendant’s claim of innocence. It only says that innocence is plausible because guilt has not been established beyond a reasonable doubt.

On the other hand, if the judge returns a guilty verdict she has concluded innocence (null) is not plausible given the evidence presented, therefore she rejects the statute of the null, innocence and concludes the alternative hypothesis- guilty.

Let’s take a closer look at how this works.

Taking a sample of 500 Penn State students, we asked them if they like cold weather, we observe a sample proportion of 0.556, since these students go to school in Pennsylvania it might generally be thought the true proportion of students who like cold weather is 0.5, in other words the NULL hypothesis is that the true population proportion equal to 0.5 ,

In order to “test” what is generally thought about these students (half of them like cold weather) we have to ask about the relationship of the data we have (from our sample) relative to the hypothesized null value. In other words, is our observed sample proportion far enough away from the 0.5 to suggest that there is evidence against the null? Translating this to statistical terms, we can think about the “how far” questions in terms of standard deviations. How many standard deviations apart would we consider to be “meaningfully different”?

What if instead of a cutoff standard deviation, we found a probability? With a null hypothesis of equal to 0.5, the alternative hypothesis is not equal to 0.50. To test this, we convert the distance between the observed value and the null value into a standardized statistic. We have worked with standardized scores when working with z scores. We also learned about the empirical rule. Combining these two concepts, we can begin to make decisions about “how far” the observed value and null hypothesis need to be to be “meaningfully different”.

To do this we calculate a Z statistic, which is a standardized score of the difference.

We can look at the results of calculating a z test (which we will do using software). Large test statistics indicate a large difference between the observed value and the null, contributing to greater evidence of a significant difference, thus casting doubt that the true population proportion is the null value.

Accompanying the magnitude of the test statistic, our software also yields a “probability”. Returning to the values of the empirical rule we know the percentiles under a standard normal curve. We can apply these to determine the probability (which is really a percentile) of getting an observed score IF the null hypothesis is indeed true (or the mean of the distribution). In this class, we will not be calculating these by hand, but we do need to understand what the “p-values'' in the output mean. In our example, after calculating a z statistic, we determine that if the true proportion is 0.5, the probability we would get a sample proportion of 0.556 is 0.0061. This is a very small probability as measure against the standard defining “small” as a probability less than .05. In this case, we would reject the null hypothesis as a probable value for the population based on the evidence from our sample.

While p values are a standard in most statistics courses and textbook there have been recent conversations about the use of p values. 

Before we proceed any further we need to step away from the jargon and understand exactly what the heck a p value is. Simply a p value is the probability of getting the observed sample statistic, given the null hypothesis is true. In our example, IF the true proportion of Penn State students who like the cold IS really .5 (as we state in the null hypothesis), what is the probability that we would get an observed sample statistic of .556?

When the probability is small we have one of two options. We can either conclude there is something wrong with our sample (however, if we followed good sampling techniques as discussed early in the notes then this is not likely) OR we can conclude that the null is probably not the true population value. 

To summarize the application of the p value:

• If our p-value is less than or equal to \(\alpha \), then there is enough evidence to reject the null hypothesis (in most cases the alpha is going to be 0.05).
• If our p-value is greater than \(\alpha \), there is not enough evidence to reject the null hypothesis.

Recall our “test” about whether Penn State students like cold weather. we have to ask about the relationship of the data we have (from our sample) relative to the hypothesized null value. In other words, is our observed sample proportion far enough away from the 0.5 to suggest that there is evidence against the null?

We can use what we know about the sampling distribution of sample proportions to help find our evidence!

Recall that under certain conditions, the sampling distribution of the sample proportion, \(\hat{p} \), is approximately normal with mean, \(p \), standard error \(\sqrt{\dfrac{p(1-p)}{n}}\), and estimated standard error \(\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}\).

In the previous section, we learned how to perform a hypothesis test for one proportion. The concepts of hypothesis testing remain constant for any hypothesis test. In these next few sections, we will present the hypothesis test for one mean. We start with our knowledge of the sampling distribution of the sample mean.

Recall that under certain conditions, the sampling distribution of the sample mean, \(\bar{x} \), is approximately normal with mean, \(\mu \), standard error \(\dfrac{\sigma}{\sqrt{n}} \), and estimated standard error \(\dfrac{s}{\sqrt{n}} \).

Notice when working with continuous data we are going to use a t statistic as opposed to the z statistic. This is due to the fact that the sample size impacts the sampling distribution and needs to be taken into account. We do this by recognizing “degrees of freedom”. We will not go into too much detail about degrees of freedom in this course.

Let’s look at an example.

In this section, we include a little more discussion about some of the issues with hypothesis tests and items to be conscious about. 

We are now going to develop the hypothesis test for the difference of two proportions for independent samples. The hypothesis test follows the same steps as one group.

These notes are going to go into a little bit of math and formulas to help demonstrate the logic behind hypothesis testing for two groups. If this starts to get a little confusion, just skim over it for a general understanding! Remember we can rely on the software to do the calculations for us, but it is good to have a basic understanding of the logic!

We will use the sampling distribution of \(\hat{p}_1-\hat{p}_2\) as we did for the confidence interval.

For a test for two proportions, we are interested in the difference between two groups. If the difference is zero, then they are not different (i.e., they are equal). Therefore, the null hypothesis will always be:

\(H_0\colon p_1-p_2=0\)

Another way to look at it is \(H_0\colon p_1=p_2\). This is worth stopping to think about. Remember, in hypothesis testing, we assume the null hypothesis is true. In this case, it means that \(p_1\) and \(p_2\) are equal. Under this assumption, then \(\hat{p}_1\) and \(\hat{p}_2\) are both estimating the same proportion. Think of this proportion as \(p^*\).

Therefore, the sampling distribution of both proportions, \(\hat{p}_1\) and \(\hat{p}_2\), will, under certain conditions, be approximately normal centered around \(p^*\), with standard error \(\sqrt{\dfrac{p^*(1-p^*)}{n_i}}\), for \(i=1, 2\).

We take this into account by finding an estimate for this \(p^*\) using the two-sample proportions. We can calculate an estimate of \(p^*\) using the following formula:

\(\hat{p}^*=\dfrac{x_1+x_2}{n_1+n_2}\)

This value is the total number in the desired categories \((x_1+x_2)\) from both samples over the total number of sampling units in the combined sample \((n_1+n_2)\).

Putting everything together, if we assume \(p_1=p_2\), then the sampling distribution of \(\hat{p}_1-\hat{p}_2\) will be approximately normal with mean 0 and standard error of \(\sqrt{p^*(1-p^*)\left(\frac{1}{n_1}+\frac{1}{n_2}\right)}\), under certain conditions.

Therefore,

\(z^*=\dfrac{(\hat{p}_1-\hat{p}_2)-0}{\sqrt{\hat{p}^*(1-\hat{p}^*)\left(\dfrac{1}{n_1}+\dfrac{1}{n_2}\right)}}\)

...will follow a standard normal distribution.

Finally, we can develop our hypothesis test for \(p_1-p_2\).

Working with two group means requires a few extra steps. We need to know the relationship between the two groups. Are the groups independent or dependent? Another important step is to understand the variability of the two groups (our old friend variability comes into play again!). If the groups have unequal variance we may have to account for that in our hypothesis test. 

Let's take a look at independence and dependence first. 

These notes will first work through independent groups and then proceed to dependent groups.