Sampling Distributions of Proportions Sampling Distribution Is the

Sampling Distributions of Proportions

Sampling Distribution • Is the distribution of possible values of a statistic from all possible samples of the same size from the same population We will use: p for the population proportion and p-hat for the sample proportion

Suppose we have a population of six people: Alice, Ben, Charles, Denise, Edward, & Frank What is the proportion of females? 1/3 What is the parameter of interest in this population? Proportion of females Draw samples of two from this population. How many different samples are possible? 6 C 2 =15

Find the 15 different samples that are possible & find the sample proportion of the number of females in each sample. Ben & Frank Alice & Ben. 5 Charles & Denise Alice & Charles. 5 Alice & Denise 1 Charles & Edward Alice & Edward. 5 Charles & Frank Alice & Frank How does the mean of the . 5 Denise & Edward p-hat) Ben & Charlessampling distribution (m 0 Denise & Frank Ben & Denise compare to the population . 5 Edward & Frank parameter (p)? m = p Ben & Edward 0 p-hat 0. 5 0 0. 5. 5 0 Find the mean & standard deviation of all p-hats.

Formulas: These are found on the formula chart!

Assumptions (Rules of Thumb) • Sample size must be less than 10% of the population (independence) • Sample size must be large enough to insure a normal approximation can be used. np > 10 & n (1 – p) > 10

Based on past experience, a bank believes that 7% of the people who receive loans will not make payments on time. The bank recently approved 200 loans. Yes – np = 200(. 07) = 14 What are the mean and standard deviation n(1 - p) = 200(. 93) = 186 of the proportion of clients in this group who may not make payments on time? Ncdf(. 10, 1 E 99, . 07, . 01804) = Are assumptions met? . 0482 What is the probability that over 10% of these clients will not make payments on time?

Assume that 30% of the students at ESH wear contacts. In a sample of 100 students, what is the probability that more than 35% of them wear contacts? mp-hat =. 3 & sp-hat =. 045826 Check assumptions! np = 100(. 3) = 30 & n(1 -p) =100(. 7) = 70 Ncdf(. 35, 1 E 99, . 3, . 045826) =. 1376