SAMPLING DISTRIBUTIONS, CENTRAL LIMIT THEOREM In this lab assignment. you will explore important properties of the sampling distribution of a sample mean in the context of a lling process. In particular, you will use some sampling procedures in R {or R commander} to demonstrate the validity of the Central Limit theorem. You will see mat the distribution of the sample mean for samples drawn from a highly skewed distribution becomes approximately normal as the sample size increases. Moreover, you will investigate how the spread of the sampling distribution of the sample mean is affected by sample size. Examining a Filling Process These days, juice box dispensing juice {such as apple juice) is performed by lling machines. These are set to deliver a certain amount of juice. which we will call the target amount. and the contents of juice boxes will vary around this mean value. The amount of variation will depend on the efciency of the machine itself as well as certain properties of thejuiec. such as its density. The manufacturer may be able to reduce this variation. but no amount of expertise or effort could lead to its complete removal. A company uses a lling machine to ll usual boxes with an applejuiee. The boxes are supposed to contain [30 milliliters (approximately 4.4 oz) of the drink. However, when buying a box of juice which bears a stamp claiming that the amount of the drink is [30 milliliters (ml), will there be exactly 130 ml of juice? Probably some amount close but not exactly equal to I30 ml. If the amount of juice dispensed by the filling machine follows a symmetric distribution and the mean target value is set equal to the claimed amount of 130 ml, half of the juice boxes would be underllcd and half would be overlled. This may seem perfectly reasonable to the manufacturer but consumers may feel differently, particularly if they happen to buy thc underlledjuicc boxes. To make the customer happy, the manufacturer may decide to overl] thejuice boxes slightly so that the target fill of the machine is more than the claimed amount. However, even a small increase in the target ll represents a loss of prot to the manufacturer. The juice boxes are shipped in packages containing either ID or 35 juice boxes. How does the amount of juice vary from juice box to juice box? How does the average amount of juice vary from package to package containing the same number of juice boxes'.' How does the number of juice boxes in a package affect the distribution of the means? You will obtain the answers to all these questions in this lab. Answer the following questions: 1. Suppose the amount of apple juice dispensed by a lling machine follows a normal distribution with a mean [p] and a standard deviation [ 0. Select the Distributions option in theR commander menu and then the Normal distribution among continuous distributions options. This allows you to obtain a graph of the normal density tnction. and to calculate normal probabilities when the parameters (p and f) are provided. Use R commander to answer the following questions: {a} Assume that the mean amount dispensed by the machine is set at u = 130 ml. Enter the value of ['- as 5, then 10, then 15, and eventually 20 m]. Aer each entry. carefully examine the shape of the corresponding density curve. You are not supposed to print the density curves. Describe briey the change in the appearance of the percentage of underlled juice boxes [the juice boxes containing less than 130 ml) when :3 decreases or increases? In general. how does the magnitude of the standard deviation affect the lling process? {b} Now assume that the mean amount dispensed by the machine is set at u = 135 ml. Enter the value of r as 15 ml. lCalculate the percentage of underlled juice boxes [the juice boxes containing less than 13-0 ml} in this case. 1l.'v'hat is the percentage of underfilledjuice boxes if F were 10 ml and 5 ml? In general. what is the effect of decreasing fon the percentage of underlled juice boxes? {c} Now set the standard deviation to 5 ml and change the mean. Enter the value of u as 130, then 135, and eventually 140 ml. Calculate the percentage of underflllcd juice boxes in each case. Describe briey how the shape of the corresponding curve changes. How does changing the value of p affect the lling process\"? Does the percentage of underlled boxes increase or decrease? Do not print the density curves. 3. Obtain the mean amount of apple juice for each sample consisting of 10 juice boxes. Make sure that all [00 columns are included in the pane] of the Wamertca! S'trmmmt'es "dialog box. {a} Obtain a frequency histogram ofthe 100 means with the bins starting at 132. ending at 14], and using a width of l. (Hint: R assumes that the right endpoint of each interval is included. Your histogram should include the left endpoints.) Paste the histogram into your report. The format of the histogram should be the same as the format of the histogram in Lab r' Instructions (labels at the axes, title}. (b) Refer to the histogram obtained in part {a}. Does the data appear to be normally distributed? Compare the distribution of the means to the distribution of individual observations studied in Question 2 in terms of their spread and degree of skewness. (c) Obtain a Q-Q plot and a boxplot for the 100 means. Add a title to each plot. Paste both plots into your report. ls (are) there any outlicrts)? Do the plots conrm your ndings in part {b}? Compare the plots with the ones in part {c} of Question 2. {(1} Obtain the sample size. mean, and standard deviation of the 100 means. Paste the summaries into your report. Compare the values with the mean and the standard deviation of the sampling distribution of the sample mean predicted by the theory of sampling distributions. What does the standard deviation mean here? Now suppose 100 packages are randomly selected, each consisting of 35 juice boxes of apple juice obtained from the population of all juice boxes lled over the same short time period. The amount of apple juice in each juice box is determined and the measurements are saved in the lab-941x: file in the form of a table of 100 columns, each consisting of 35 rows