Confidence Intervals for Proportions Chapter 19 part 1

The sampling distribution for 𝑝 is centered at 𝑝 Recall from chapter 18: The sampling distribution for 𝑝 is centered at 𝑝 and the standard deviation of the sampling distribution is 𝑝𝑞 𝑛 . We use 𝑝 to represent our sample-based estimate of the true proportion of a population.

Marine scientists say that 10% of the world’s reef systems have been destroyed in recent times. At the current rate, 70% of the reefs could be gone in 40 years. One reason for this decline is disease. In a sample of 104 sea fans from Las Redes Reef, 54 were found to be infected with aspergillosis. What is the sample proportion of infected sea fans? 𝑝 = 54 104 =0.519 How can we use this to say something about the true population proportion? Let’s find out…

For the sea fans, we have (0.519)(0.481) 104 =0.049 𝑜𝑟 4.9% When we estimate the standard deviation of a sampling distribution, we call it a standard error and we have to use 𝑝 𝑎𝑛𝑑 𝑞 . 𝑆𝐸 𝑝 = 𝑝 𝑞 𝑛 For the sea fans, we have (0.519)(0.481) 104 =0.049 𝑜𝑟 4.9%

This means there is a 95% chance that 𝑝 is no more than 2SE from 𝑝 . Confidence Intervals Because this is a Normal model (always check the conditions/assumptions) we know that 95% of all samples will have a proportion within 2 standard errors of center. This means there is a 95% chance that 𝑝 is no more than 2SE from 𝑝 . 𝑝 +2𝑆𝐸=0.519+2 0.049 =0.617 𝑝 −2𝑆𝐸=0.519−2 0.049 =0.421 Conclusion: We are 95% confident that between 42.1% and 61.7% of Las Redes sea fans are infected.

What does it mean when we say we have 95% confidence that … ? It means that 95% of samples of the same size will produce confidence intervals that capture the true proportion.

𝑝 ±2𝑆𝐸 We found our confidence interval by calculating 𝑝 ±2𝑆𝐸. The 2SE part is called the margin of error. 𝑝 ±2𝑆𝐸 estimate margin of error

Today’s Assignment HW: page 455 #3-4