Imagine you are in a game show.  There are 10 prizes hidden on a game board with 100 spaces.  One prize is worth $50, two are worth $20, and another seven are worth $10.  You have to pay $5 to the host if your choice is not correct.  Let the random variable x be the winning. (a)  (1 pts) Complete the following probability distribution. (Show the probability in fraction format and explain your work) x P(x) -$5 $10 $20 $50 (b)  What if your expected winning or loss in this game? Be specific in your answer whether it’s winning or loss. (Show work and round the answer to two decimal places) (c)   What is the standard deviation of the probability distribution? (Show work and round the answer to two decimal places)

Discussion C: Central Limit TheoremWeek 5 Post (300+ words):Collect some quantitative data (if your data from week 1 is quantitative, you can use it). Find the sample mean and standard deviation. Plot it in a histogram. Does the data seem to follow the bell curve of the normal distribution? What features of the data do or do not fit in with the shape of the normal curve. How much deviation from the curve is to be expected?Now perform a normality test on your data (Shapiro-Wilk test: http://sdittami.altervista.org/shapirotest/ShapiroTest.html or http://www.brianreedpowers.com/MAT240/stats/descriptiveStats.html)– the test will give you a p-value. The higher the p-value, the more closely your data follows the normal distribution. Based on the test, do you think your data could have been drawn from a normal distribution?Week 6 Responses (100+ words x2):Choose two of your classmates’ data sets. Take 30 random samples of 5 data points each (one way: Past the data here http://www.randomizelist.com/ randomize the list and take the first 5 numbers, or use the sampling feature at http://www.brianreedpowers.com/MAT240/stats/descriptiveStats.html), and calculate the average for each of these samples. You will now have 30 sample means. Create and post a histogram for your sample means. What is the mean of these means? What is the standard deviation? Does this make sense based on the Central Limit Theorem? Do the sample means follow a normal distribution? What p-value does the normality test give? How and why does this differ from the original data?REPLY MY CLASSMATE: Jasmine AliceaThe quantitative data I chose to use was the number hours I slept on an average week, as discussed from my previous post. Finding the mean (or average) of my numbers, simply list all variables in ascending order à 6.5,7,7,7,7.5,8,9. Add all the variable to equal one sum (=52). Then divide by the number of numbers presented. 52/7= 7.428. We want to present the sum to the nearest hundredth; to do so, we need to round up (7.43). To find the median, we need find the middle number between all variables. Fortunately, I have 7 variables and can find the median between the first three numbers and last three numbers (7).Standard DeviationEquation for standard Deviation for sample is as follows:Sample Deviationà Sample(x)=n−1∑∣x−xˉ∣2 = Square root of unbiased sample variancePopulation Deviation equationà σ=N∑(xi−μ)2∑= “sum of”X= is a value in the data setμ=is the mean of the data setN= is the number of data points in the population (not used for this week’s discussion N-1= is used for dealing with a sample variable. (we are using in this week’s discussion)To find the sample subtract the variable by the mean times square- Add them together and then divide by mean.S-(squared)(6.5-7) sq+(7-7)sq+(7-7)sq+(7-7)sq+(7.5-7)sq+(8-7)sq+(9-7)sq7-1n-1 (mean)0.25+0+0+0.25+1+4= 5.5/6= à0.93 Unbiased sample varianceS= sq root of unbiased sample variance à 0.93 squared = 0.86 Sample standard deviationFrequency Histogram4321>=0 2 4 6 8 10I’m not sure if the histogram follows the bell curve of a normal distribution, as I am still trying to see how all the different representations equal the original data; hours I sleep per week. If I am analyzing it visually, I can see how the average time in hours I sleep will be 7, as the spike in the bell curve is narrow and in between 6-9. It does represent a normative distribution.The deviation from the curve is skewed to the right. If my calculations are correct, I sleep on average 7 hours and 45mins but on a good night’s rest 8 ½ hours, and a not-so-great night, 6 hours.The Shapiro-Wilk test is usually recommended for data with <10 variables. It provides a Null Hypothesis (sample size of n comes from a normal distribution).Results: Shapiro-Wilk Normality Testn = 7Mean = 7.428571428571428SD = 0.8380817098475258W = 0.885157374718934Threshold (p=0.01) = 0.7300000190734863 --> HO acceptedThreshold (p=0.05) = 0.8029999732971191 --> HO acceptedThreshold (p=0.10) = 0.8379999995231628 --> HO accepted--> “Your data seems normal”REPLY TO MY CLASSMATE:Tabitha ThuguriThe central limit theorem states that the sampling distribution of any statistic will be normal or nearly normal, if the sample size is large enough. Requirements for accuracy.The more closely the sampling distribution needs to resemble a normal distribution, the more sample points will be required. Requirements for accuracy.The more closely the sampling distribution needs to resemble a normal distribution, the more sample points will be required.

Small boxes of NutraFlakes cereal are labeled “net weight 10 ounces.” Each hour, random samples of size n = 4 boxes are weighed to check process control. Five hours of observations yielded the following: Weight Time Box 1 Box 2 Box 3 Box 4 9 A.M. 9.8 10.4 9.9 10.3 10 A.M. 10.1 10.2 9.9 9.8 11 A.M. 9.9 10.5 10.3 10.1 Noon 9.7 9.8 10.3 10.2 1 P.M. 9.7 10.1 9.9 9.9 Using these data, construct limits for x- and R- charts. Is the process in control? What other steps should the QC department follow at this point?