Sampling Distributions of Proportions A Quick Review

Sampling Distribution Is the distribution of possible values of a statistic from all possible samples of the same size from the same population In the case of the pennies, it’s the distribution of all possible sample proportions (p) We will use: p for the population proportion and p-hat for the sample proportion

What is the proportion of females? Suppose we have a population of six people: Alice, Ben, Charles, Denise, Edward, & Frank What is the proportion of females? What is the parameter of interest in this population?   Draw samples of two from this population. How many different samples are possible? 1/3 gender 6C2 =15

These are found on the formula chart! Formulas: These are found on the formula chart!

Does the standard deviation of the sampling distribution equal the equation? So – in order to calculate the standard deviation of the sampling distribution, we MUST be sure that our sample size is less than 10% of the population! WHY? We are sampling more than 10% of our population!

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

Remember back to binomial distributions Why does the second assumption insure an approximate normal distribution? Remember back to binomial distributions Suppose n = 10 & p = 0.1 (probability of a success), a histogram of this distribution is strongly skewed right! np > 10 & n(1-p) > 10 insures that the sample size is large enough to have a normal approximation! Now use n = 100 & p = 0.1 (Now np > 10!) While the histogram is still strongly skewed right – look what happens to the tail!

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. What are the mean and standard deviation of the proportion of clients in this group who may not make payments on time? Are assumptions met? What is the probability that over 10% of these clients will not make payments on time? Yes – np = 200(.07) = 14 n(1 - p) = 200(.93) = 186 Ncdf(.10, 1E99, .07, .01804) = .0482

Suppose one student tossed a coin 200 times and found only 42% heads Suppose one student tossed a coin 200 times and found only 42% heads. Do you believe that this is likely to happen? np = 200(.42) = 84 & n(1-p) = 200(.58) = 116 Since both > 10, I can use a normal curve! Find m & s using the formulas. No – since there is approximately a 1% chance of this happening, I do not believe the student did this.

Assume that 30% of the students at CHS wear contacts Assume that 30% of the students at CHS wear contacts. In a sample of 100 students, what is the probability that more than 35% of them wear contacts? Check assumptions! mp-hat = .3 & sp-hat = .045826 np = 100(.3) = 30 & n(1-p) =100(.7) = 70 Ncdf(.35, 1E99, .3, .045826) = .1376