Binomial sampling and the binomial distribution project Instructions: Part 1- Binomial Distribution in Sampling Data Collection 1. Create a model of a population consisting of individuals of two different types and a method of selecting a simple random sample of 10 individuals from your population. In your model, you'll need to know ahead of time how many individuals there are of the two types in your population. Your population should consist of at least five of each type of individual, and should have at least 200 individuals total. For example: o you could have your population consist of at least 200 grains of rice, some of which have been dyed with food coloring (at least five dyed and five not dyed). A sample can be chosen by mixing the rice together in a bag and reaching in without looking to pull out 10 grains of rice. you could let your population be represented as a column of values O and 1 in Excel (at least 200 values total, including at least five O's and five 1's). A random sample can be chosen by creating a second column adjacent to the one containing your population values and entering the command "=RAND( in each cell in the new column to create a random sequence of numbers. Next, sort this new column. To do so, find the "Sort & Filter" menu in the Editing section of the toolbar in Excel, then click "Sort Smallest to Largest". Doing so will automatically rearrange the rows in both columns so that the new column is in increasing order. This means that the first column will now be in a random order. so a simple random sample can be taken by reading the first several values in the reordered column of Os and 1s. Here's a demonstration of this procedure. o you could invent some other model and sample selection method meetine the above criteria 2. Choose at least 30 samples (of size 10) from your model population and record the number of each type of individual within each sample. For this, simply designate one type as a success and the other as a failure, and count the number of successes in your sample (so you don't also have to count the failures. since this can alwavs be determined from the number of successes). Analysis/Questions 1. Create a frequency table of the number of successes in your samples. Make your classes consist of the individual values from 0 to 10 rather than grouping into larger classes. This way, your table shows exactly how many samples had O successes, 1 success, and so on. 2. Create a histogram showing the information in your frequency table. 3. How many individuals were present in your population altogether? How many successes were present in your population altogether? 4. If represents the number of successes in a random sample of size 10 from your population, then the distribution of is approximatelv binomial According to theory, what is the expected (or mean) number of successes in a single sample? According to theory, what is the standard deviation of the number of successes in a single sample? 5. Find the mean and (sample) standard deviation of your 30+ values of X. Are they close to the theoretical mean and standard deviation? 6. What percentage of our samples had values within one (theoretical) standard deviation of the (theoretical) mean! What percentage of your samples had values within two (theoretical) standard deviations of the (theoretical) mean? Instructions: Part 2- Central Limit Theorem Data Collection 1. Use the applet found here B to explore the Central Limit Theorem. First click the button labelled "Normal" to specify that you want to sample from a normally distributed population. A graph of a normal distribution will appear, and its mean p and standard deviation o will be displayed. Record the mean and standard deviation. 2. The applet is designed to select many samples randomly from your specified population and plot an approximate histogram of the mean values of the samples taken. Click 'n=2" to see the histogram of sample means when the samples have size 2. The mean ( and standard deviation oz are displayed as well. Record these values. Click the remaining sample sizes 'n=2", 'n=9;, and so on, observing the shape of the distribution of sample means and recording the mean Mg and standard deviation 0 each time. 3. Click 'New Dist' to change the distribution of the population. Choose either "Skewed Left' or 'Skewed Right!. Record the mean p and standard deviation a for this new distribution. 4. Repeat step 2 above for this new distribution Analysis/Questions 1. For steps 1 and 2, when the population is normally distributed, what did you observe about the shape of the distribution of sample means as the sample size n was increased from 2 to 100? 2. For steps 3 and 4, when the population is skewed, what did you observe about the shape of the distribution of sample means as the sample size n was increased from 2 to 1002 3. What did you observe about the mean of the sample means u compared to the mean of the underlying population p? What relation between u and u; is predicted by the Central Limit Theorem? Are your observations consistent with the Central Limit Theorem's predictions? 4. What did you observe about the standard deviation of the sample means a= compared to the standard deviation of the underlying population o? What relation between o and o- is predicted by the Central Limit Theorem? Are your observations consistent with the Central Limit Theorem's predictions? Show an explicit calculation of the predicted value of o, for at least one value or n