5.1 Sampling Distributions for Counts and Proportions. Ulrich Hoensch MAT210 Rocky Mountain College Billings, MT 59102

Size: px

Start display at page:

Download "5.1 Sampling Distributions for Counts and Proportions. Ulrich Hoensch MAT210 Rocky Mountain College Billings, MT 59102"

Horace Shelton
5 years ago
Views:

1 5.1 Sampling Distributions for Counts and Proportions Ulrich Hoensch MAT210 Rocky Mountain College Billings, MT 59102

2 Sampling and Population Distributions

3 Example: Count of People with Bachelor s Degrees The proportion of people in the U.S. with a Bachelor s degree is = 22.6%. A SRS of 4 people is taken; let X be the random variable that describes the number of people in the sample who have a Bachelor s degree. Find each probability: None of the 4 people in the sample has a Bachelor s degree. P(X = 0) = P(NB, NB, NB, NB) = ( ) 4 = (0.774) 4 = %. Exactly 1 of the 4 people in the sample has a Bachelor s degree. P(X = 1) = P(B, NB, NB, NB) + P(NB, B, NB, NB) +P(NB, NB, B, NB) + P(NB, NB, NB, B) = (0.226)(0.774) 3 + (0.774)(0.226)(0.774) 2 +(0.774) 2 (0.226)(0.774) + (0.774) 3 (0.226) = 4(0.226)(0.774) 3 = %.

4 Example: Count of People with Bachelor s Degrees Exactly 2 of the 4 people in the sample have a Bachelor s degree. P(X = 2) = P(B, B, NB, NB) + P(B, NB, B, NB) +P(B, NB, NB, B) + P(NB, B, B, NB) +P(NB, B, NB, B) + P(NB, NB, B, B) = 6(0.226) 2 (0.774) 2 = %. Exactly 3 of the 4 people in the sample have a Bachelor s degree. P(X = 3) = 4(0.226) 3 (0.774) = %. All 4 people in the sample have a Bachelor s degree. P(X = 4) = (0.226) 4 = %.

5 Example: Count of People with Bachelor s Degrees The probability distribution for the number X of people in the SRS of size n = 4 is consequently: k P(X = k) We can compute these probabilities using the calculator as follows. Using the binompdf() command (in the DISTR menu), we have: P(X = 0) = binompdf(4, 0.226, 0) = ; P(X = 1) = binompdf(4, 0.226, 1) = ;... P(X = 4) = binompdf(4, 0.226, 4) = In this situation, the random variable X is binomially distributed with n = 4 observations and probability of success p = We write X B(4, 0.226).

6 ios App Use Go to Distribution Calculator at the bottom of the main menu and click on Binomial Distribution.

7 ios App Use

8 Binomial Setting and Distributions

9 Sampling Distribution of a Count

10 Example: Count of People with Bachelor s Degrees Given that the count X of people with a Bachelor s degree in a SRS of size 100 has the distribution X B(100, 0.226), find these probabilities. There are exactly 20 people with a Bachelor s degree in the sample. P(X = 20) = binompdf(100, 0.226, 20) = %. There are at most 20 people with a Bachelor s degree in the sample. Here we could compute P(X = 0), P(X = 1),..., P(X = 20) separately and then add them. It is easier to use binomcdf() on the calculator. P(X 20) = binomcdf(100, 0.226, 20) = %.

11 Example: Count of People with Bachelor s Degrees There are more than 20 people with a Bachelor s degree in the sample. Here we could compute P(X = 21), P(X = 22),..., P(X = 100) separately and then add them. It is easier to realize that and so we get P(X > 20) = 1 P(X 20), P(X > 20) = 1 binomcdf(100, 0.226, 20) = %.

12 ios App Use

13 ios App Use

14 Binomial Mean and Standard Deviation Since the count X is a discrete random variable, its mean and standard deviation can be computed using the formulas in Section 4.4. When this is done in general, we obtain the following formulas.

15 Example: Count of People with Bachelor s Degrees If a SRS of 100 people is taken, then the mean of the number X of people in the sample with a bachelor s degree is and the standard deviation is µ X = np = = 22.6, σ X = np(1 p) =

16 Example: Chytrid Fungus in Frogs Suppose the population proportion of mountain yellow-legged frogs who test positive for chytrid fungus is p = 0.60 = 60%. Suppose now that a survey involving 50 frogs is conducted. Find each probability: 1. The probability that in the sample, the sample proportion ˆp of frogs with the fungus is greater than 80%. 2. The middle 95% of the sampling distribution of ˆp. Instead of using the binomial distribution, we use a normal approximation, as explained in the following.

17 Mean and Standard Deviation of a Sample Proportion Example The mean and the standard deviation in the previous example are: µˆp = p = 0.60, p(1 p) (0.60)(0.40) σˆp = = = %. n 50

18 Normal Approximation for Counts and Proportions

19 Normal Approximation for Counts and Proportions

20 Example: Chytrid Fungus in Frogs As computed above, µˆp = 0.60, σˆp = So, ˆp N(0.60, 0.07) (approximately). 1. The probability that the sample proportion is greater than 80% is P( ˆp > 0.80) = normalcdf(0.80, , 0.60, 0.07) = %. 2. The cutoffs for the middle 95% of the sampling distribution of ˆp are: the lower cutoff is given by invnorm(0.025, 0.6, 0.07) = %; the upper cutoff is given by invnorm(0.975, 0.6, 0.07) = %.

21 ios App Use

Binomial and Normal Distributions. Example: Determine whether the following experiments are binomial experiments. Explain.

Binomial and Normal Distributions Objective 1: Determining if an Experiment is a Binomial Experiment For an experiment to be considered a binomial experiment, four things must hold: 1. The experiment is