Stat200 Lab

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Feb 20, 2024

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9/2/22 © Pennsylvania State University LAB 3.2 STAT 200: Lab Activity for Section 3.2 Understanding and Interpreting Confidence Intervals – Learning Objectives: • Identify the parameter being estimated and the sample statistic that gives the best estimate • Construct a confidence interval for a parameter based on a sample statistic and margin of error • Use a confidence interval to recognize plausible values of the population parameter • Construct a confidence interval for a parameter given a sample statistic and an estimate of the standard error • Interpret (in context) what a confidence interval says about a population parameter Activity 1: Sample from a population to build a confidence interval The virtual bead box represents voters in a medium-size town. The color of each bead represents each individual’s opinion (for or against) on a ballot proposition about property taxes. Assume the election is in one week and we are interested in estimating support for the proposition. Let yellow beads be in support and green beads be against. Make sure you calculate the statistic for Yellow beads. The website provides for the green beads. In this activity we will learn about the people’s opinions by taking a sample from the population and building a confidence interval for the proportion of town voters in favor of the ballot proposition. 1. Which parameter are we interested in? Use proper notation! 2. Take a sample from the bead box and use your sample to calculate the appropriate sample statistic for the yellow beads. Use proper notation! 3. It is known that the standard error for this sample statistic is 0.0837. Use this fact to build a 95% confidence interval for the parameter. 4. Interpret your confidence interval in context . Refer to lecture slides if needed. We are 95% [blank1] that the [blank2] proportion of voters in favor of the proposition is between [blank3] & [blank 4]. 5. Based on your confidence interval, can you say with 95% certainty that the budget proposition will pass on Election Day? - yellow beads All the beads P a beads ↑ = 5 = - 66 · 6612(0 . 0837)(4926 , 08274) We are 45 % confident that the true proportion of voters in favor of the Proposition is between · 4926 and 8274 Yes , it will because the number is pretty high

9/2/22 © Pennsylvania State University LAB 3.2 Activity 2: Building confidence intervals for Hollywood movie budgets A sampling distribution is shown for budgets (in millions of dollars) of all movies to come out of Hollywood in 2011, using samples of size n = 20. We see that the standard error is about 10.232. For each of the sample means listed below, a. Find their location in the sampling distribution in the figure above. b. Use the standard error of 10.232 to find the 95% confidence interval to estimate the population mean c. Also, indicate whether the confidence interval successfully captures the true population mean of 53.481 million dollars. 1. ࠵?̅ = 40 2. ࠵?̅ = 70 3. ࠵?̅ = 84 0 -- 40 I2(10 . 232) (19 . 536 , 60 . 464) Yes , it does 70 12(10 . 232) (49 . 336 , 90 464) Yes , it does 8412(10 . 232)(63 . 336 , 104 . 464) No , it does not

9/2/22 © Pennsylvania State University LAB 3.2 Activity 3: Sample size and confidence intervals For this activity, we will use the students in all STAT 200 classes in a semester as our population. The data set “ GPA .csv” is the file containing responses to the student survey. We will use this population to investigate the relationship between sample size, standard error, and width of confidence intervals. Let’s consider the variable GPA. In StatKey, upload the data set into the sampling distribution for one mean option, and select the correct variable GPA. Report all quantities to two decimal places to make the calculations friendlier. 1. Using a sample size of n=10, generate a sampling distribution with at least 5000 samples. a. Where is the center of this sampling distribution? b. What is the standard error? c. If we were to build a 95% confidence interval using one of the sample means, how wide would it be? 2. Now using a sample of size n=50, generate a sampling distribution with 5000 samples. a. Where is the center of this sampling distribution? b. What is the standard error? c. If we were to build a 95% confidence interval using one of the sample means, how wide would it be? 3. Finally, using a sample of size n=100, generate a sampling distribution with 5000 samples. a. Where is the center of this sampling distribution? b. What is the standard error? c. If we were to build a 95% confidence interval using one of the sample means, how wide would it be? 4. What happens to the center of the distribution as the sample size increases? 3 . 256 0 . 141 3 . 003 I2(0 . 141) = (2 . 721 , 3 . 285) 3 258 0 862 3 35212(0 . 062) = (3 228 , 3 - 476) 3 . 259 0 043 3 : 3 =2(0 043) (3 214 , 3 . 386) The Center stays the same

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9/2/22 © Pennsylvania State University LAB 3.2 5. What happens to the standard error, and the width of confidence intervals as the sample size increases? Activity 4: Interpreting a confidence interval. [Page 222 in our textbook is helpful.] Using a sample of 24 deliveries described in “Diary of a Pizza Girl” on the Slice website, we find a 95% confidence interval for the mean tip given for a pizza delivery to be $2.18 to $3.90. Which of the following is a correct interpretation of this interval? 1. I am 95% sure that all pizza delivery tips will be between $2.18 and $3.90. 2. 95% of all pizza delivery tips will be between $2.18 and $3.90. 3. I am 95% sure that the mean pizza delivery tip for this sample will be between $2.18 and $3.90. 4. I am 95% sure that the mean tip for all pizza deliveries in this area will be between $2.18 and $3.90. 5. I am 95% sure that the confidence interval for the mean pizza delivery tip will be between $2.18 and $3.90. Activity 5: Election Day 1. The results are in! The proposition passed with 69.6% of the vote. Did your confidence interval contain the true value for the population parameter? Note that ‘yes’ and ‘no’ are both right here, as long as it accurately reflects your confidence interval. 2. Assuming your instructor has 600 students that will perform this experiment today, each using a sample size of 30. How many of the students do you expect to have calculated confidence intervals that contain the true value for the population parameter, at 95% Confidence? The standard arrow decreased The wide of confidence interval decreased = 0 & Yes -- - 600(95) = 570

Stat200 Lab

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