Most people are right-handed and even the right eye is dominant for most people. Researchers have long believed that late-stage human embryos tend to turn their heads to the right. German bio-psychologist Onur Güntürkün (Nature, 2003) conjectured that this tendency to turn to the right manifests itself in other ways as well, so he studied kissing couples to see whether both people tend to lean their heads to the right more often than to their left (and if so, how strong the tendency is). He and his researchers observed couples from age 13 to 70 in public places such as airports, train stations, beaches, and parks in the United States, Germany, and Turkey. The observers were careful not to include couples who were holding objects such as luggage that might have affected which direction they turned. We will assume these observations are representative of the overall decision-making process when kissing.

 

We could employ a “trial-and-error” type of approach to determine which values of 𝜋 appear plausible based on what we observed in the sample. This involves testing different values of 𝜋 and seeing whether the corresponding two-sided p-value is larger than some pre-specified cut-off, typically 0.05. (This cut-off is often called the level of significance.) That is, we will consider 𝜋₀ a plausible value for 𝜋 if assuming 𝜋 = 𝜋₀ does not make our sample statistic look surprising (yielding a small p-value).

Use the One Proportion Inference applet to determine the values of 𝜋0 such that observing 80 of 124 successes or a result more extreme occurs in at least 5% of samples. Hints: Enter the sample size and number of successes (or proportion) from the study. Check the Two-sided box and use the pull-down menu to specify "Tail probabilities."  Also use the Exact Binomial p-values. Use values of 𝜋0 that are multiples of 0.01 until you can find the boundaries where the exact two-sided p-values (using the tails of the binomial distribution) change from below 0.05 to above 0.05. Then feel free to “zoom in” to three decimal places of accuracy if you’d like. You can check the Show sliders box in the applet and use the slider or edit the orange number to change the value of 𝜋. Keep in mind that you are changing the conjectured value of 𝜋, not the observed number of successes, which should stay at 80.

 

The algorithm we have used here, finding the values so the sum of the (smaller) tail probabilities is larger than the level of significance is sometimes called the "smallest tail probability" approach (aka Blaker's method).

Clopper-Pearson Method: Alternatively, another way to define a binomial confidence interval is to find all the values of 𝜋 such that P(X  ≤ observed) < (significance level)/2 and P(X ≥ observed) < (significance level)/2. 

Definition: A confidence interval (CI) specifies the plausible values of the parameter based on the sample result.

What you found in (g) and (i) will be called a “95% confidence interval” as it was derived using the 1 −0.95 = 0.05 cut-off value/significance level.  And what you found in (h) would be a 99% confidence interval. (So 99% is the level of confidence.)

With more decimal places, you should have found the first 95% confidence interval to be (0.557, 0.727).

You could interpret the interval by saying you are 95% confident that the probability a kissing couple leans to the right is between 0.557 and 0.727, meaning that we are 95% confident that, in the long-run, between 55.7% and 72.7% of kissing couples lean right.

Different software packages will calculate confidence intervals differently, but if the sample size is large, you shouldn't see too much difference between the methods.

More important than the choice of methods is how to interpret the intervals and their properties.

Discussion: When the sample size is large, you won't see much difference between the different algorithms. So focus more on how to interpret the produced intervals.  You should also be able to predict how changing the sample size and or changing the confidence level (e.g., 95% vs. 99%) changes the interval.

Also keep in mind the likely duality between confidence intervals and tests of significance: The confidence interval is the set of values for which we would fail to reject the null hypothesis in favor of the two-sided alternative for a particular level of significance (e.g., confidence level = (1 - significance level)).