PART I - Confidence Intervals
For the first part of the project I would like you to create a confidence interval (90% CI) for the population proportion (p) of all soda drinkers in Western New York who prefer Pepsi.
Remember the purpose of a confidence interval is to estimate a value of the population (either the population mean or the population proportion as is the case here). We only have a sampling of 30 willing participants here. I am sure that your surveys were convenience based and thus we definitely do not want to assume that your result is the actual percentage of all soda drinkers in Western New York who prefer Pepsi. So...let's try to make a statistically sound statement based on our sample data using confidence intervals.
In a Separate Word document type your answers the following:
1. What is your point estimate for the population proportion?
2. What is the critical value for a 90% Confidence Interval of the population proportion? Is your Critical Value a Z-score or a T-score? How did you determine which to choose?
3. What is the margin of error (the numeric value you calculated)?
4. Using steps 1-3, what is the 90% confidence interval for the population proportion of Western New Yorkers who prefer Pepsi?
5. In your own words explain what the 90% confidence interval tells you.
6. If we increased our level of confidence to 99% what would you expect to find? Why?
PART II - Hypothesis Testing
For the second part of the project I would like you to test the hypothesis that more Americans prefer Pepsi products to Coke products (this is my own bias hypothesis!). So test the hypothesis (at a 5% level of significance) that more than 50% (greater than 0.50) of Western new Yorkers prefer Pepsi products.
Check the notes for hypothesis testing (using the Classical Approach) to recall that you first need to determine your Null and Alternate Hypothesis. In our case the claim (that more than 50% prefer Pepsi) is an alternate hypothesis (since it is only greater than and not greater than or equal to). Next determine the critical values for the hypothesis test based on the direction of the test (alternate hypothesis) and the level of significance.
To determine the critical value you will need to open the SurfStat Applet and choose the correct picture on the "standard normal" applet (**remember** the alternate hypothesis is greater than we have a right tailed test) and enter 0.05 for the probability. This will give you the critical value or "cut-off" point for the rejection zone.
Now you need to determine the test statistic. Recall that the test statistic is always computed based on the sample data. See your notes for the specific formula for the test statistic of a proportion. Never round off your steps - always wait to round until you are finished with your computations. If you choose to round within a problem be sure to round to at least 4 decimal places. You will need to use your sample percentage for those who preferred Pepsi in the test statistic formula. Remember that the "hypothesized percentage (or p)" is 0.50. So for the numerator of the test statistic you should be subtracting 0.50 and your sample percentage of those who prefer Pepsi (if your percentage for those who prefer Pepsi was greater than 50% than the test statistic will be positive, if your percentage for those who prefer Pepsi is less than 50% than the test statistic will be negative). Don't forget to divide by the denominator of the formula - remember that p stands for the hypothesized mean (0.50 in this case) and q stands for the complement (which will also be 0.50 in this case)
To test a hypothesis with the Classical Approach you need to compare the critical value and the test statistic. If the test statistic falls within the critical region (greater than the critical value in our case since we have a right tailed test) we will REJECT the NULL HYPOTHESIS. If the test statistic falls outside the critical region (less than the critical value in our case since we have a right tailed test) we will FAIL TO REJECT the NULL HYPOTHESIS.
Now for our conclusion:
IF we choose to REJECT the NULL HYPOTHESIS we will conclude that there IS enough evidence at the specified level of significance (there is always enough evidence when you reject - that is a strong verdict) to either accept or reject the said claim (you must decide which is correct here - accept or reject). To make this determination simply look at the original claim - if it was an alternate hypothesis choose "accept" ("a" for alternate and "a" for accept) and if the original claim was a null hypothesis choose the word reject (this should make sense since if you choose REJECT the NULL HYPOTHESIS and the Null IS indeed your claim you are therefore rejecting your claim!)
IF we choose to FAIL to REJECT the NULL HYPOTHESIS we will conclude that there is NOT enough evidence at the specified level of significance (there is always NOT enough evidence when you FAIL to reject - like a not guilty verdict) to either accept or reject the said claim (you must decide which is correct here - accept or reject). To make this determination simply look at the original claim - if it was an alternate hypothesis choose "accept" ("a" for alternate and "a" for accept) and if the original claim was a null hypothesis choose the word reject (this should make sense since if you choose FAIL to REJECT the NULL HYPOTHESIS and the Null IS indeed your claim you are therefore stating that there is not enough evidence to reject your claim!)
In a Word document type your answers the following:
7. How did we determine that this was a right tailed test?
8. What was your critical value? In what case would you have 2 critical values?
9. What was your test statistic? How many formulas are there for test statistics? How do you determine which to use for a specific problem?
10. What was your verdict? Did you decide to REJECT the NULL HYPOTHESIS or FAIL to REJECT the NULL HYPOTHESIS? How did you make that determination?
11. Which of the following would be your concluding statement: a. There is enough evidence at the 5% level of significance to accept the claim that more than 50% of Americans prefer Pepsi products. b. There is enough evidence at the 5% level of significance to reject the claim that more than 50% of Americans prefer Pepsi products. c. There is not enough evidence at the 5% level of significance to accept the claim that more than 50% of Americans prefer Pepsi products. d. There is not enough evidence at the 5% level of significance to reject the claim that more than 50% of Americans prefer Pepsi products.
12. Explain in your own words how you arrived at the choice above. In other words, why did you choose either a, b, c, or d?