Section 7 Lab Activity

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Clemson University *

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3090

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Statistics

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Apr 3, 2024

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STAT 3090 S ECTION 7 L AB S PRING 2024 S AMPLING D ISTRIBUTIONS N AME : Cam Allen O BJECTIVES : Upon successful completion of this assignment, you will be able to…  Recognize and use different sampling methods  Describe the distribution of the sample mean and the sample proportion  Understand the Central Limit Theorem and its application  Understand the impact of Sample size on Sampling Variability  Interpret the probability of sampling distribution Part One: Sampling Techniques 1. A Beary Interesting Problem (18 points) To determine the effectiveness of jazz music in terms of bear attack prevention, a few students want to set up a study where they casually approach 7 bears while playing jazz music and then approach 7 bears with no music playing. Thus, they will need a sample of 14 bears. The students are near a wilderness area with 6 regions and they are also near to a zoo that has 5 bears. The 6 wilderness regions are approximately the same size and the number of bears in each area is approximately the same. Explain how the students could use each of the different types of sampling methods to gather their data. Note that some of the sampling methods would be nearly impossible to do. That is ok. Simply come up with a way they could use that particular method even if it is completely impractical. (Have fun with this) Disclaimer: Please do not try this experiment. a. Simple random sampling (3 pts) They could create a list of all available bears between the wild and zoo ones, then randomly select them. b. Stratified random sampling (3 pts) Separate the bears into two groups, wilderness and zoo. Then randomly select 7 from each group. c. Cluster sampling (3 pts) Create 2 clusters to group the wilderness bears into, then randomly pick one of the clusters and sample all bears in that cluster. d. Systematic sampling (3 pts) Randomly select 1 bear, then sample each 7 th bear after that first one. e. Convenience sampling (3 pts) Sample the first 14 bears they see or find. 1

STAT 3090 S ECTION 7 L AB S PRING 2024 S AMPLING D ISTRIBUTIONS f. Judgement sampling (3 pts) Choose the 14 bears they think best represent the overall bear population. Sampling Distribution of Sample Mean 2. Credit Card Debt (10 points) The mean credit card debt for a U.S. household is $7,115 with a standard deviation of $2,160. This mean is such a large value because of a few deeply indebted households. Consider a random sample of 50 U.S. households and let x represent the sample mean credit card debt. Use this information to answer the following questions. (5 pts each) (a) Describe the sampling distribution of sample mean x , i.e., its shape (normal or not), mean, and standard error. (Keep 2 decimal places.) Shape is normal, standard error is 305.47. (b) What is the probability that the mean credit card debt for a sample of 50 U.S. households is less than $6,500? Include probability notations and round your answer to 4 decimal places. P(x < 6500) = 0.0220 3. Hot Dog Fat Content (15 points) A hot dog manufacturer claims that its new lean beef hot dogs have an average of 3.5g of fat per hot dog. The fat content of these hot dogs is known to be approximately normally distributed with a standard deviation of 0.5g. (5 pts each) (a) What is the probability that one of these lean beef hot dogs has more than 4g of fat? Include probability notations and round your answer to 4 decimal places. P(X > 4) = 0.1587 (b) Describe the sampling distribution of the average fat content of a randomly selected package of 8 lean beef hot dogs, i.e., shape, mean, and standard error. Round to two decimal places if necessary. 2

STAT 3090 S ECTION 7 L AB S PRING 2024 S AMPLING D ISTRIBUTIONS Symmetrical shape, bell curve. Mean is 3.5, while standard error is 0.18. (c) What is the probability that a randomly selected package of 8 lean beef hot dogs will have an average fat content greater than 4g? Use 4 decimal places in your final answer of the required probability and include probability notations. P(Z > 2.828) = 0.0023 4. WebAssign #1, 2, 3, 4, 8 (17 points) ****************************************************************************** Part 2: Sampling Distributions of sample proportion 5. College Degrees (9 points) A survey is conducted from a population of people of whom 40% have a college degree. Of the 60 people who were surveyed, 27 indicated that they have a college degree. (a) What is the sample proportion, p , of respondents who have a college degree in this survey? Include the correct symbol for the sample proportion in your answer. (4pts) p = 27/60 = 0.45 (b) Find the probability of obtaining a sample proportion less than or equal to the one that you calculated in part (a) from a sample of size 60. (5pts) To get full credit, you need to:  Use the sampling distribution of sample proportion p .  Determine the shape, mean, and standard error of p .  Draw a sketch with appropriate area shaded.  Use the proper probability notations.  Get the correct answer with 4 decimal places. P( p < 0.45) = P (Z < 0.45-0.4/sqrt(0.4*0.6/60)) = P(Z < 0.79) = 0.7852 3

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STAT 3090 S ECTION 7 L AB S PRING 2024 S AMPLING D ISTRIBUTIONS 6. Baking Shows (8 points) A nationwide survey analyzing trends in popular media found that 81% of U.S. college students prefer British baking shows over American baking shows. You are interested to see if this result holds at your university. You take a sample of 140 students on campus and find that 125 of them prefer watching British baking shows. (a) Will the sampling distribution of sample proportion, p , of students at your university who prefer British baking shows follow a normal distribution? Check the appropriate conditions to explain your answer. (3 points) The distribution will follow a normal distribution because it is independently drawn and expected counts are greater than 10. (b) What is the probability of obtaining a sample of size 140 where 125 or more students prefer British baking shows? Include proper probability notations and round your answer to 4 decimal places. (5 points) P(p ̅> 125/140) = P(Z > 2.50) = 0.0062 7. Park Visitors (8 points) Leslie has been tasked with putting together a statistical report regarding the use of a particular park in her hometown. Previous data shows that 72% of the residents living in the town visited the park in the last month. She wants to know the probability that more than 98 individuals in a random sample of 150 residents have visited the park in the last month. (a) Will the sampling distribution of sample proportion, p , of residents who have visited the park in the last month follow a normal distribution? Check the appropriate conditions to justify your answer. (2 points) Yes, expected counts are greater than 10. 4

STAT 3090 S ECTION 7 L AB S PRING 2024 S AMPLING D ISTRIBUTIONS (b) What is the lower bound you will use for p for Leslie’s sample of 150 residents to solve the problem? Round your answer to two decimal places. (2 points) The lower bound I will use is 0.65. (c) What is the probability that at least 98 individuals in a random sample of 150 residents have visited the park in the last month? Include proper probability notations and round your answer to 4 decimal places. Use the sampling distribution of p to find the probability. (4 points) P(X>98) = P( p > 0.65) = 0.9719 8. WebAssign #5, 6, 7, 9 (15 points) 5

Section 7 Lab Activity

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