Properties of a sample proportion
Sample proportions vary from sample to sample. Our model (of sampling from an underlying population) therefore means that they have probability distributions.
The sample proportion in a particular category from a random sample of size n has a distribution that ...
Count and proportion of successes
Although the sample proportion in a category, p , is a good summary statistic, the raw count of sample values in the category, x = np, contains equivalent information and is often easier to use.
Both p and x = np have distributions with same shape (other than the scaling constant n).
Brand loyalty
Brand awareness and brand loyalty are extremely important marketing tools, and many advertising campaigns focus on the brand name rather than on specific products. The power of a recognised brand name is shown by the following experiment.
Two ninth grade students at Burnaby South Secondary School in Vancouver conducted the following science project. Sixteen test subjects were blindfolded, given a slice of pizza that they were told was ordered from Pizza Hut and another slice that they were told was ordered from a local pizza delivery shop, and were asked to pick which slice tasted better. Unknown to the subjects, all slices were purchased from the same source, but all 16 test subjects picked the Pizza Hut slice as 'best'.
We will investigate this type of experiment with a simulation. If the brand name had not influenced decisions on which slice was best, the probability that any subject picked the 'Pizza Hut' slice as best would be 0.5 and, providing the subjects did not talk to each other about their choices, the choices would be independent.
The diagram below performs this simulation — values are sampled from an infinite categorical population of a half 'Pizza Hut' and a half 'Other' values.
Click the checkbox Accumulate then take a few samples to build up the distribution of the proportion picking Pizza Hut as best in samples of size n = 16. The distribution is discrete since only a few distinct values are possible. The distribution of the sample proportion should be centred on π = 0.5.
Use the pop-up menu to change the sample size and repeat sampling to verify that the sample proportion has lower spread when the sample size is higher.
Finally, use the pop-up menu to change the scale on the sampling distribution of the proportions to show the the distribution of the sample number of times that Pizza Hut is chosen as best. Observe that the shape of the distribution is unchanged.