Section 7 Lab - Megan McCauley

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Clemson University *

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3090

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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 1 N AME : Megan McCauley 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) Assign a number to every bear acrpss all 6 regions and the zoo, then put all the numbers on a spinning wheel and spin the wheel 14 times to get 14 bears. b. Stratified random sampling (3 pts) The two strata are the bears from the wilderness regions and the bears from the zoo. Randomly select 1 bear from each wilderness region, then select 3 additional bears across all the wilderness regions, and include all 5 zoo bears in the study. c. Cluster sampling (3 pts) The clusters are each individual wilderness region and the zoo; there are 7 clusters total. Randomly select two bears from each cluster. d. Systematic sampling (3 pts) Let’s say there are 23 total bears in the wilderness regions. 23+5 zoo bears is 28. 28/14=2, which is the sampling interval. Select every 2 nd bear until 14 bears have been picked. e. Convenience sampling (3 pts) The students would use the 5 bears from the zoo and then 9 more bears in the region closest to them.
STAT 3090 S ECTION 7 L AB S PRING 2024 S AMPLING D ISTRIBUTIONS 2 f. Judgement sampling (3 pts) The students could go to the zoo and to each of the 6 regions and pick the 14 bears that they feel are most representative of the entire population by using metrics like age and color of fur. 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 𝑥̅ 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 , i.e., its shape (normal or not), mean, and standard error. (Keep 2 decimal places.) Mean: 2160 Standard deviation: 2160 Shape: normal Standard error = 𝜎 √𝑛 = 2160 √50 =305.18 N≥30 so normal (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. Z= (6500−7115) ( 2160 √50 ) = -2.017 𝑃(𝑧 < −2.017) = 0.0228 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)
STAT 3090 S ECTION 7 L AB S PRING 2024 S AMPLING D ISTRIBUTIONS 3 (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. 𝑧 = (𝑥−𝜇) 𝜎 = (4−3.5) 0.5 = 1 𝑃(𝑧 > 1) = 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. Shape: the sample size is less than 30, so the shape is abnormal Mean: 3.5g Standard error = 𝜎 √𝑛 = 0.5 √8 = 0.177 = 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. 𝑧 = (𝑥−𝜇) 𝜎 = (4−3.5) 0.177 = 2.825 𝑃(𝑧 > 2.825) = 0.0024 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, 𝑝̅ , 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 𝑝̅ .
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