University:
High School
Course:
High School Physics
Academic year:

2020
Views:

372

Pages:

2
Author:

Addison Munoz

Central Limit Theorem When we are looking at sample means, their distribution will be approximately Normal, no matter what shape the population was. A number that describes the whole population is known as a parameter. A number that is calculated from a sample is known as a statistic. We always use statistics to estimate parameter Sampling Distributions for Means and Proportions Proportions Means (categorical data) (quantitative data) Parameter 𝑝 𝜇 Statistics 𝑝̂ 𝑥̅ Mean 𝑝 𝜇 St. Deviation 𝑝𝑞 √ 𝑛 𝜎 What conditions must be satisfied? Formula for a z-score 𝑧= 𝑣𝑎𝑙𝑢𝑒 − 𝑚𝑒𝑎𝑛 𝑠𝑡. 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 √𝑛 𝑧= 𝑣𝑎𝑙𝑢𝑒 − 𝑚𝑒𝑎𝑛 𝑠𝑡. 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 Large Counts 𝑛𝑝 ≥ 10 𝑛𝑝 ≥ 10 Population Distribution is Normal or 𝑛 ≥ 25 𝑏𝑦 𝐶𝐿𝑇 < 10% 𝑛 ≤ 10% 𝑜𝑓 𝑁 𝑛 ≤ 10% 𝑜𝑓 𝑁 Example: According to Greenpeace, 80% of all pollination is performed by bees. Biologists across the country randomly select a total of 60 plants to be monitored by camera to determine how they are pollinated. What is the probability that less than 75% of the monitored plants are pollinated by bees? Example: The mean life span of a brand name tire is 50,000 miles. Assume that the life spans of the tires are normally distributed, and the population standard deviation is 800 miles. If you select 100 tires, what is the probability that their mean life span is more than 50,200 miles? 𝜇 = 50000 𝑝 = 0.8 𝑞 = 0.2 𝜎 = 800 𝑛 = 100 𝑛 = 60 Sampling distribution has the mean of 𝜇𝑝̂ = 0.8 and the standard deviation of 𝜎𝑝̂ = √ 0.8∙0.2 60 Sampling distribution has the mean of 𝜇𝑥̅ = 50000 800 and the standard deviation of 𝜎𝑥̅ = = 80 √100 = 0.05 Conditions: Conditions: 50000 𝑡𝑖𝑟𝑒𝑠 < 10% 𝑜𝑓 𝑎𝑙𝑙 𝑚𝑎𝑛𝑢𝑓𝑎𝑐𝑡𝑢𝑟𝑒𝑑 𝑡𝑖𝑟𝑒𝑠 60 𝑝𝑙𝑎𝑛𝑡𝑠 < 10% 𝑜𝑓 𝑎𝑙𝑙 𝑝𝑙𝑎𝑛𝑡𝑠 100 ≥ 25 𝑙𝑎𝑟𝑔𝑒 𝑠𝑎𝑚𝑝𝑙𝑒 𝑛𝑝 = 60 ∙ 0.8 = 48 ≥ 10 𝑛𝑞 = 60 ∙ 0.2 = 12 ≥ 10 The conditions are met, we can use the Normal model 𝑁(50000, 80) The conditions are met, we can use the Normal model 𝑁(0.8, 0.05) 50200 − 50000 = 2.5 80 𝑃(𝑧 ≥ 2.5) = 0.006 𝑧= 0.75 − 0.8 = −1 0.05 𝑃(𝑧 ≤ −1) = 0.1587 𝑧= The probability that less than 75% of the monitored plants are pollinated by bees is 0.1587. The probability that the mean life span of 100 tires is more than 50,200 miles is 0.006.