Chapter 19 Week 9, Monday

Introductory Example You just graduated with a Statistics degree and started your first job at Petco. Your official title is “Data Analyst” and you work within the Marketing department.

Introductory Example You just graduated with a Statistics degree and started your first job at Petco. Your official title is “Data Analyst” and you work within the Marketing department. Petco is considering opening a new store in Akron, and your team has been put in charge of determining whether or not this move will bring profits to the firm. There are thousands of factors that go into this analysis. For example, what proportion of Akron dog owners will purchase an outfit for their dog within the next year?

Introductory Example Consider a representative sample of 250 Akron dog owners. When asked the question “do you plan to purchase an outfit for your dog within the next year”, 15% of the respondents said yes. Question: What can we say about the proportion of Akron dog owners as a whole? (in other words, what can we say about p) Answer: As long as certain assumptions are met about our sample, we can say a lot. For example: there is a 95% chance that between 10.6% and 19.4% of Akron dog owners plan to buy an outfit within the next year.

Goals for this Chapter 1.Define the assumptions that must be met about the sample 2.Define the formulas for finding the “confidence interval” 3.Learn how (and how not) to interpret confidence intervals 4.Practice calculating confidence intervals and stating meaningful conclusions.

The Assumptions 1.Sample must be randomly selected, with independent data values. 2.The sample size (n) must be smaller than 10% of the population size (N). 3.Both and must exceed 10.

The Formulas Calculate the “standard error of p-hat”: Calculate the “margin of error”: State the “confidence interval”: Confidence LevelZ* 90% % %2.576

The Interpretation Recall the dog example. Our sample proportion was 15%. The margin of error (although we did not state this explicitly) was 4.4%. Therefore our 95% confidence level was 15% ± 4.4%. Interpretation #1: “15% of all Akron dog owners will buy an outfit for their pet in the next year”

The Interpretation Recall the dog example. Our sample proportion was 15%. The margin of error (although we did not state this explicitly) was 4.4%. Therefore our 95% confidence level was 15% ± 4.4%. Interpretation #2: “It is probably true that 15% of all Akron dog owners will buy an outfit for their pet in the next year”

The Interpretation Recall the dog example. Our sample proportion was 15%. The margin of error (although we did not state this explicitly) was 4.4%. Therefore our 95% confidence level was 15% ± 4.4%. Interpretation #3: “We don’t know exactly what proportion of dog owners will buy an outfit for their pet in the next year, but we know that between 10.6% and 19.4% will.”

The Interpretation Recall the dog example. Our sample proportion was 15%. The margin of error (although we did not state this explicitly) was 4.4%. Therefore our 95% confidence level was 15% ± 4.4%. Interpretation #4: “We don’t know exactly what proportion of dog owners will buy an outfit for their pet in the next year, but the interval 10.6% and 19.4% probably contains it.” TRUE; But we can do better

The Interpretation Recall the dog example. Our sample proportion was 15%. The margin of error (although we did not state this explicitly) was 4.4%. Therefore our 95% confidence level was 15% ± 4.4%. Interpretation #5: “We don’t know exactly what proportion of dog owners will buy an outfit for their pet in the next year, we are 95% sure that the interval 10.6% and 19.4% contains it.” This is an awesome answer

Back to our Petco Example Recall:n=250 (sample size) (sample proportion) At 90%, our confidence interval is: 11.3% and 18.7% At 95%, our confidence interval is: 10.6% and 19.4% At 99%, our confidence interval is: 9.2% and 20.8% “For a given sample size, if you increase the confidence level, you increase the margin of error”

Back to our Petco Example Assume:95% Confidence Level (sample proportion) At n=100, our confidence interval is: 8% and 22% At n=250, our confidence interval is: 10.6% and 19.4% At n=3000, our confidence interval is: 13.7% and 16.3% “For a given confidence level, if you increase the sample size, you decrease the margin of error”