Part 1: What does it mean to be 95% confident? Before we get started, it's very important to think carefully about the general format of the confidence interval. This format is: sample statistic + margin of error The sample statistic is always directly in the center of the interval since it is used to construct the interval, and the margin of error determines the width of the interval. 1. In your own words, explain what is meant by the term "sample statistic." In other words, what is a sample statistic?We now want you to use a program called StatKey. This is a free online program that you can find here: http://www.lock5stat.com/StatKey/index.html In the row labeled Sampling Distributions, click on the link for Proportion. You will be taken to a tool that will allow you to draw samples and examine how the confidence intervals change for each sample. For this part of our lab activity, we'll be focusing on the proportion of college graduates in the United States. According to the US Census Bureau, 27.5% (or 0.275 if expressed as a proportion) of adults who are at least 25 years old have a college bachelor's degree or higher. Note that on the right side of the screen, you can see some information (as shown below) about our population. We will be taking many random samples from this population in order to learn more about how confidence intervals behave. Data Tables Confidence Intervals Original Population Proportion 0.275 2. Click on the button that says "Generate 1 sample." This will simulate taking a random sample of 200 individuals from the population of adults who are least 25 years old. In your sample of size n = 200, what is the sample proportion? This number appears on the right side of the screen, in a table that gives you a Count (i.e., the number of individuals out of 200 who have a college bachelor's degree or higher), the Sample Size (i.e., 200), and the Proportion (i.e., the Count divided by the Sample Size). Write your sample proportion down below. 3. Click the "Generate 1 sample" button again. What is your new sample proportion?Click the "Generate 1 sample" button at least 10 more times. Note that each time you generate a new sample, the sample proportion is plotted as a black dot in the "Sampling Dotplot of Proportion" on the left side of the screen. 4. Why do you think the sample proportions are different each time you select a sample? Now, in the area on the right side of the page, click on the tab that says "Confidence Intervals." When you do this, you'll see a black vertical line right at 0.275. This represents the population proportion. You will also see several horizontal lines. Each line has a circle right in the center that represents the sample proportion; the lines on either side of the circle represent the margin of error that has been added and subtracted to the sample proportion. If the lines are green, you will notice that they cross the vertical black line (meaning those intervals include the true population proportion). If the lines are red, they do not cross the vertical and so do not include the true population proportion. 5. Click now on the Reset Plot button that appears above the Sampling Dotlplot of Proportion. Then, click again on the "Generate 1 sample" button. On the right side of the screen, make sure to click the Confidence Intervals tab. Look carefully at the line that appears in the middle portion of the screen. If you hover your mouse over the circle that appears directly in the middle of the line, you will see the sample proportion. A. What is the sample proportion you are seeing for the sample you just generated? Write the value of that sample proportion below. B. The point where the line begins is the lower bound of the confidence interval, and the point where the line ends is the upper bound of the confidence interval. As you look along the x-axis, what are the approximate values of the lower and upper bounds of the confidence interval? C. Does the true population proportion of 0.275 fall somewhere between the lower and upper bounds of the confidence interval you generated to answer part (B)?D. Are you surprised that the true population parameter either does or does not fall between the lower and upper bounds of your confidence interval? Please explain. E. Would you expect everyone in STAT 1350 to have obtained the exact same confidence interval that you obtained? Please explain why or why not. 6. Click again on the Reset Plot button. Next, click on the "Generate 100 samples" button to simulate drawing 100 more samples. Each sample, again, consists of 200 adults who are 25 years or older. Look now at all the confidence intervals that have been constructed based on each sample statistic. Near the top of this list of intervals you'll see the word "Coverage," with a percentage below it. As you might guess, this number is the percentage of all of the confidence intervals that have been generated that contain the population proportion of 0.275. A. What is the percentage you are seeing? B. How do you think that percentage should be interpreted? In other words, what is it telling us? 7. Click again on the "Generate 100 samples" button again to simulate drawing 100 more samples. Now, look again at all of the confidence intervals that have been generated. Under "Coverage," what is the new percentage?The percentages you wrote above in responses to Questions 6 and 7 might not be equal to exactly 95%, but hopefully, they will be close to 95%, and if we draw more and more samples and construct more 95% confidence intervals, we'd expect that number to be approximately 95%. The reason we can be 95% confident that any single interval will include the true population parameter is that we know, over the long run, if we could construct many 95% confidence intervals, 95% of them would include the true population parameter and 5% would not. Put differently, we can be 95% confident that our one interval will include the population parameter because we know we constructed the interval using a method that gets things right 95% of the time. Look again at what you did for Questions 6 and 7. In particular, look carefully at the many confidence intervals that were constructed each time you generated 100 samples. Based on what you see, please answer Questions 8 and 9. 8. TRUE or FALSE: The population parameter will always be between the lower and upper bounds of the confidence interval. 9. TRUE or FALSE: The sample statistic will always be between the lower and upper bounds of the confidence interval. Part 2: Practice constructing a 95% confidence interval You want to estimate the proportion of all OSU students who plan to enroll in summer courses. You survey 340 randomly selected OSU students and find that 83 of these students plan to enroll in summer courses. 10. What would your sample proportion be? Please round that proportion to three decimal places. p = 1 1. Is the proportion you wrote in your answer to Question 10 a parameter or a statistic? Please explain.12. Is the true population proportion of all OSU students who plan to enroll in summer courses a parameter or a statistic? Please explain. Recall the general format of the confidence interval. This format is: sample statistic + margin of error When we construct a confidence interval based on a sample proportion, the format of our interval, using appropriate symbols, is as follows: p tz' P(1 -p) n Remember that p is the sample proportion, n is our sample size, and z* is a standardized value (see below) that corresponds to the level of confidence. As you can see from the formula above, although you had initially learned about a "quick estimate" for the margin of error in Chapter 3 (where you simply determined -), there is a way for us to determine a more precise margin of error. When working with confidence intervals for proportions, the margin of error is determined based on this formula: z*( ). The z* values based on the different confidence levels are given below. 90% 95% 99% 1.64 1.96 2.58 13. From what you see above as you look at the formula to compute the margin of error, what three things affect the size of the margin of error? 14. Use your sample proportion (from Question 10) to construct a 95% confidence interval. As you are calculating the margin of error, round your work to three decimal places in 6order to be as precise as possible, and show all of your work! In the spaces below, write down the lower and upper bounds of your confidence interval. Lower bound: Upper bound: 15. Refer back to your work in Question 14. Write a statement below in which you express your confidence. Begin the statement with "I am 95% confident that..."