Problem: Can we conduct a null hypothesis test (normal distribution, and a 33.3% claim) to determine both if three brands of Hershey's products are equally popular (1), and if Reese's is more popular (2) than the other two products?
Observed Data:
$n = 450$ //Total # people who bought Hershey's products (i.e., sample size)
- Total: 450
- Reese's (R): 173 ($\frac{173}{n} = 38.44\%$)
- Kisses (K): 149 ($\frac{149}{n} = 33.11\%$)
- Whoppers (W): 128 ($\frac{128}{n} = 28.44\%$)
$p = 33.33\%$ //Claim Percentage (i.e., equal share among products)
$\sigma = \sqrt{\frac{p(1 - p)}{n}} = \sqrt{\frac{0.3333(1 - 0.3333)}{450}} = 2.22\%$ //Standard deviation
We then plot the above values to a normal distribution chart (claim percentage being the mean, and then +- 2 standard deviations) -- you can view the chart image here: https://i.stack.imgur.com/8VeOc.jpg
From this we can see that only Kisses (K) falls within the likely 95% region, and so we can conclude that the three products are not equally popular with customers (at a 5% level of significance).
Obviously it appears as if Reese's are more popular, however the question is whether we can actually make any determination from this about Reese's being more popular than the other two brands. Can you test more than one thing at a time? How would we test whether one product is more popular than two others? Wouldn't we need a different claim percentage? If so, what might that be?
Sure you can, using a Chi Square Test you can test for equality of each proportion where your Null Hypothesis would be Ho: P1 = P2 =...Pk (however many proportions your using, in this case 3), so your Null Hypothesis becomes Ho: P1 = P2 = P3. Now, your Alternative Hypothesis would be something like "At least one proportion is different" (you can be more specific than this). I would have commented this rather than posting as an "answer" because its more like a hint, but my reputation is so low I cannot make comments yet.
I'm going to assume you know a little about the Chi Square, but if you don't just do a simple google search..You already have your "Observed Counts" which is the Reeses = 173, Kisses = 149 and Whoppers = 128.. The 33% is very important because this is the percentage we use to calculate the "Expected Counts". You just multiply the total sample size by this, so 450*(1/3) = 150. So of 450 samples you would "expect" about 150 for each...Using these 2 values: "Observed Count" and "Expected Count", you can plug them into the Chi Square Formula and eventually obtain a p-value.. I don't have enough experience here to post the formulas super nice for you, but I will leave that to you to figure out :p
If you don't have statistical software such as "R" or a graphing calculator.. this is a useful link.. http://www.chisquarecalculator.com/