Section 7 Lab Activity

STAT 3090 S ECTION 7 L AB F ALL 2023 S AMPLING D ISTRIBUTIONS N AME : 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) Create a system for assigning a unique number to each bear, and then, by generating a random list of numbers, choose 'n' bears. b. Stratified random sampling (3 pts) 1

STAT 3090 S ECTION 7 L AB F ALL 2023 S AMPLING D ISTRIBUTIONS Gather a straightforward random sample from both the regions and the zoo, ensuring that these samples are representative of the region's population size. c. Cluster sampling (3 pts) Choose one of the six regions or the zoo at random, and survey all the bears in that selected area. d. Systematic sampling (3 pts) Approach the first bear you come across and then survey every fifth bear you encounter thereafter. e. Convenience sampling (3 pts) Examining the 5 bears within the zoo may not yield an accurate sample due to the possibility of zoo animals being more docile or accustomed to human presence. f. Judgement sampling (3 pts) You suggest using only the bears from region 1, given 2

STAT 3090 S ECTION 7 L AB F ALL 2023 S AMPLING D ISTRIBUTIONS shape, mean and standard error for the sampling distribution of the mean fat content of 8 lean beef hot dogs?) Shape: since the distribution of the fat content is approximately normally distributed, then the sampling distribution of is approx. normal ? Center: ?? µ = µx = x̄ 3.5 g? Variability: ?? σ =σx/√n = 0.5/√8 = 0.18 g x̄ ?? ? (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? For this problem find the z-score and include 4 decimal places in your final answer of the required probability. P ( x̄ > 4) = P( Z > (4 ― 3.5/0.18) ) = P(Z > 2.78) = 0.0027 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 of respondents who have a college degree. Include the correct symbol for the sample proportion in your answer. (4pts) p̂ = 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). Use a sketch to help you solve the problem and use proper probability notation. Keep four decimal places. (5pts) Shape: Since np= 60(0.4) = 24 ≥ 5 and n(1 ―p ) = 60(1 ― 0.4) = 36 ≥ 5, the sampling distribution ?? ? ? of is approx normal with p̂ Center: µ ?? p̂ = = 0.4 p p? Variability: σ ?? p̂ = √p(1 ―p/n ) = √(0.4(1 ― 0.4) /60) = 0.0632 ??? ? ? Then, 4

STAT 3090 S ECTION 7 L AB F ALL 2023 S AMPLING D ISTRIBUTIONS P ? (? p̂ ≤ 0.45) = ( ≤ (0.45 ― 0.4/0.0632) ) = ( ≤ 0.7911) = 0.7856 P? Z? P? Z? 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. (c) Can you use the normal distribution to approximate probabilities for the sample proportion ^ p of students at your university who prefer British baking shows? Check the appropriate conditions to explain your answer. (5 points) Yes. Since np= 140(0.81) = 113.4 ≥ 5 and (1 ― p ) = 140(0.19) = 26.6 ≥ 5, we can say that is approximatly normal p̂ (d) What is the probability of obtaining a sample where ^ p is at least as great as your sample proportion. Would this cause you to suspect that the national result is an overestimate or an underestimate for your university? (3 points) 7. Park Visitors (8 points) 5

STAT 3090 S ECTION 7 L AB F ALL 2023 S AMPLING D ISTRIBUTIONS 8. WebAssign #5, 6, 7, 9 (15 points) 7