Drawing Conclusions from Examples

The hypothesis

The very first thing a researcher (that's you!) does - before the experiment and certainly before analyzing the experimental data - is come up with a hypothesis. This is a very specific prediction about what we might expect the data to show. It is important to set up the hypothesis before we analyze the data since this will determine the ways in which we look at the data.

Often, the hypothesis will be defined for you in the question, but it is still useful to keep it in mind throughout the analysis and conclusion.

The experiment and analysis

As per the example, say we collected the following data:

Since the data is grouped, we can plot each of the sets of data using a histogram.

Histogram 1, StudySmarter Originals

Histogram 2, StudySmarter Originals

From the graphs, we can see that the mode class for the sunny data is the $60\le x<70$ class, whereas for the non-sunny weather the mode class is $0\le x\le 15$.

We can also find the mean as an additional measure of central tendency. Recall that to find the mean from grouped data, we need to use the midpoints of each class.

Now we can find the mean for the 'sunny days' revenue:

$\overline{r_1}=\frac{\sum _{}\mathrm{m}\times f_1}{\sum _{}{f}_1}=\frac{15+67.5+595+1305+3135+4030+2400}{5+17+29+57+62+30}=\frac{11547.7}{200}=57.74$ to 2 d.p.

And now find the mean for the 'non-sunny days' revenue:

$\overline{r_1}=\frac{\sum _{}\mathrm{m}\times f_2}{\sum _{}{f}_2}=\frac{390+1102.5+1015+1215+935+520+0}{52+43+35+27+17+8+0}=\frac{5252.5}{182}=28.86$ to 2 d.p.

The conclusion

So, we have collected and analyzed the data. All that is left to do is to compare our statistics with the hypothesis we made beforehand.

Recall: the revenue of the lemonade stand is higher when the sun is out.

Now we compare our original hypothesis with the statistics we have found. We can see that the average amount of revenue on a sunny day is $59.95, while the average revenue on a day that is not sunny is $18.24. Since the sunny-day revenue is so much higher than otherwise, we can conclude that the data we have collected supports our hypothesis and that, according to the data, the revenue of the lemonade stand is higher when the sun is out.

This process - drawing a conclusion about a population based on results collected from a sample - is called statistical inference.

But beware! It is important to be careful of the language we use in our hypothesis. Take for example the following statement: "sunny weather makes people more likely to buy lemonade". While this may be true, we don't know enough from the data itself to confirm that it was the weather specifically that inspired more people to buy lemonade. Instead, it could have been the case that the weather increased the number of potential customers in the vicinity of the lemonade stand.

We can also use other statistics to add to our conclusion. We can see that the range of the 'sunny day' revenues is 90 whereas the range of the 'not sunny days' is 70. Therefore we can also add to our conclusion that while there is on average a greater amount of revenue on sunny days, there is also a greater range in the revenue.