Download Pointers -‐‑Section 8.2

Pointers -­‐‑Section 8.2
Goal: To understand the concepts of a sampling distribution of sample proportions. Review: A sampling distribution of a statistic is a probability distribution for all possible values for that statistic for a sample size of size n.* (A sampling distribution is created by taking many, many samples from the same population.) A sampling distribution for the sample proportion, 𝑝, is the probability distribution for all possible values of the random variable 𝑝 computed from a sample of size n from a population with population proportion p.* (Take many, many sample from the population, calculated 𝑝 from each sample. All of 𝑝’s create the sampling distribution for the sample proportion.) Notation. • The population proportion is represented by the letter p. (There is ONLY ONE p – population proportion.) • The sample proportion is represented by the symbol 𝑝. Read: “p-­‐‑hat.” #
o 𝑝 = , where x is the number of individuals in a sample with a specified $
characteristic. o There are many, many 𝑝’s – sample proportions. • Example: In a sample of 320 children, 221 say that vanilla is their favorite flavor of ice cream. %%&
o 𝑝 =
= 0.69 '%(
o Think: 69% of children in the sample prefer vanilla ice cream. Central Limit Theorem for 𝒑 − Sampling Distribution of 𝒑 For a simple random sample of size n with population p: • The shape of the sampling distribution of 𝑝 is approximately normal provided: 𝑛𝑝 1 − 𝑝 ≥ 10 • The mean of the sampling distribution of 𝒑 is 𝜇4 = 𝑝 • The standard deviation of the sampling distribution of 𝒑 is: 𝑝 1−𝑝
𝑛
NOTE: 𝜎4 is also called the standard error for 𝒑. 𝜎4 =
WHY IS THIS IMPORTANT? If you know you have a normal distribution, you can use all of the concepts discussed in chapter 7 using the mean and standard deviation calculated above. Developed by Sharleen McCarroll in conjunction with the textbook Interactive Statistics: Informed Decisions Using Data, by Michael Sullivan III & George Woodbury, 2016. th
Reference: Sullivan III, Michael, Fundamentals of Statistics: Informed Decisions Using Data, 4 ed. Pearson, 2014.