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Math 1040 Skittles Term ProjectChanghun KimMath 104012/10/17Report IntroductionThe goal of this project is to compare date collected by the class for 2.17oz bags of Skittles. I was asked to count each color and report the totals to my teacher who then compiled the data. Once the data was all collected I applied concepts learned in class to provide the charts and figures below. Data CollectionEach student in the class will purchase one 2.17-ounce bag of Original Skittles and record the following data:Number of red candiesNumber of orange candiesNumber of yellow candiesNumber of green candiesNumber of purple candiestotal1110131211570.1930.175.228.211.1931Entire class results:RedOrangeYellowGreenPurpleTotal33228637027031215700.2110.1820.2360.1720.1991First, each member of the class purchased a 2.17-ounce bag of Original Skittles. I searched far and wide, and after going to several grocery stores. I counted the number of red, orange, yellow, green and purple candies form the bag. My bag of skittles had 11 red, 10 orange, 13 yellow, 12 green, 11 purple skittles. Using the data compiled from the entire class, record the following information: The total number of candies in the sample = 57 Organizing and Displaying Categorical Data: ColorsI recorded the proportion of each color from the sample data gathered from the class. I created a Pie Chart and a Pareto Chart for the numbers of candies of each color. To create the pie chart for this data, I listed the color categories in the first five cells of columnOrganizing and Displaying Quantitative Data: the Number of Candies per BagColumnnMeanStd.devMinQ1MedianQ3MaxTotal2660.3853.8174759616365The distribution of the histogram is skewed to the right and doesn’t appear to have a symmetrical bell shape to it due to the skewed look. The graphs that are shown above were what I expected to see. With the histogram, it shows how many times numbers over 60 appear in the graph. The boxplot shows the 5 number summary of the skittles collected. Comparing my skittles collection to the class’s collection, the class collection has a right normal distribution making it a bit different from my skittles.ReflectionCategorical data is data that is collected with numbers that don’t have a special meaning to them. Some examples would be social security numbers and colors. Categorical data deals more with names than numbers. Quantitative data is data that is collected with numbers that have meaning to them. An example would include time. Quantitative data deals with a finite and infinite numbers. For the calculations, the pie chart and pareto chart both make sense for the categorical data because they deal with putting colors together. Having a histogram for categorical data wouldn’t make sense because a histogram deals with numbers and the amount of times a number appears in a data set. For quantitative data, the histogram and boxplot make sense because they show the number values. A pie chart wouldn’t work because showing the numbers in a pie chart doesn’t look right.Confidence Interval EstimatesA confidence interval is a range (or an interval) of values used to estimate the true value of a population parameter. The purpose is to find out a true proportion of a sampleConstruct a 99% confidence interval estimate for the true proportion of yellow candies.Construct a 95% confidence interval estimate for the true mean number of candies per bag.Construct a 98% confidence interval estimate for the standard deviation of the number of candies per bag.With a 98% confidence interval level, the repeated experiment for the proportion of candies per bag, skittles would have a confidence interval estimate for the standard deviation such as intervals contain the estimated population mean would be 98%.Hypothesis TestsHypothesis testing refers to the formal procedures used in statistical analysis to accept or reject statistical hypotheses. A statistical hypothesis is an assumption about a population parameter. This assumption may or may not be true. The usual process of hypothesis testing consists of several steps. A basic outline is as follows:? Formulate the null hypothesis (HO) and the alternate hypothesis (H1).? Identify a test statistic that can be used to assess the truth of the null hypothesis.? Draw a graph to include the test statistic, critical values, and critical region (if usingthe critical value method).? Reject the null hypothesis (HO) if the test statistic is in the critical region. Fail to reject the null hypothesis if the test statistic is not in the critical region.? Restate this previous decision in simple, non-technical terms, and address the original claim.Use a 0.05 significance level to test the claim that 20% of all Skittles candies are red.Use a 0.01 significance level to test the claim that the mean number of candies in a bag of Skittles is 55.The first hypothesis test we had to do was to test the claim that 20% of all skittle candies are red. With the data collected from our sample, our proportion of red skittles came to 21.1%. After calculating the critical values with a 0.05 significance level and the z-score , we found that a 20% proportion of red skittles is very plausible and therefore we failed to reject the claim. Below are the calculations for the hypothesis test.The second hypothesis test we were to complete was to test the claim that the mean number of candies in a bag of skittles is 55. As discussed in the second confidence interval estimate we did, our sample mean was 60.385 and with a 95% confidence interval we determined that the true mean was between 58.843 and 61.927. With that information we had a pretty good idea that we would be rejecting the claim. By calculating the critical test statistic with a 0.01 significance level we found that in order for that claim to be true, our t value must fall between -2.787 and 2.787. In reality, the t value came out to be 7.194, which is way outside the range of acceptable numbers, therefore, we reject the claim that the mean number of candies in a bag of skittles is 55. ReflectionTo get accurate statistics while determining an interval estimate and preforming a hypothesis test, there are certain requirements that must be met. For constructing a confidence interval estimate for a population proportion the requirements are as follows; he sample is a simple random sample, the conditions for the binomial distribution are satisfied, there are at least five successes and at least 5 failures. For constructing a confidence interval estimate for a population mean the requirements include; the sample is a simple random sample, and the population is normally distributed or n>30. For constructing a confidence interval estimate for a population standard deviation the requirements are; the sample is a simple random sample, and the population must have normally distributed values even if the sample is large. For testing a claim about a population proportion the requirements are; the sample observations are a simple random sample, the conditions for a binomial distribution are satisfied, and the conditions np>/=5 and nq>/=5 are both satisfied. For testing a claim abut a population mean the requirements are; the sample is a simple random sample, and the population is normally distributed or n>30. For those requiring conditions for a binomial distribution to be satisfied, that means that there is a fixed number of independent trials having constant probabilities and each trial has to outcome categories of success or failure. One possible error that occurs from using this data has to do with our sample size. We only used 25 bags of skittles and one of the requirements, because our sample is not normally distributed, is that our sample size needs to be greater than 30. Another possible error could occur from inaccurate information, whether the data was recorded wrong, or some students got the wrong size bag of skittles. This sample could be improved by using a larger sample size, and also by verifying the data submitted. Students could count the skittles in class and have another classmate double check their work to be sure the data was submitted correctly. It is very interesting and helpful to see all how these math problems are applicable to real life. This seems like a very effective way to determine the quality control of a product a manufacture is supplying. It is also helpful to see how a consumer can verify that they are getting the right amount of product they are paying for. ................
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