2. Goodness-of-fit tests Normal population Manufacturing processes, such as coin
ID: 3249577 • Letter: 2
Question
2. Goodness-of-fit tests Normal population Manufacturing processes, such as coin minting, are subject to small variations due to variations in materials, temperature, and humidity. The variations in materials, temperature, and humidity from their norms are just as likely to be positive as negative but are more likely to be small than large. Consider the € 1 coin issued by Belgium. Realizing that there are small variations in the minting process and random error in the weighing process, it might be reasonable to assume that the population of coin weights is normally distributed. Let's confront this assumption with sample data and see how it fares. A random sample of 250 Belgian € 1 coins was selected. Each of the 250 coins was weighed and its weight (in grams) recorded. The sample meankis 7.520 grams, and the sample standard deviation s is 0.036 grams. The questions that follow walk you through the steps of a test of the hypothesis that the population of weights of Belgian € 1 coins has a normal distribution with a mean of 7.520 grams (the sample mean) and a standard deviation of 0.036 grams (the sample standard deviation). Note that you are using the sample mean as an estimate of the population mean and the sample standard deviation as an estimate of the population standard deviation. [Data source: A sample of size n 250 was randomly selected from the sample of size n 2,000 in the Joumal of Statistics Education data archive, euroweigh dat data set.] Mean 7.53 Standard Deviation 0.040 7.35 7.45 7.50 7.55 7.60 7.65Explanation / Answer
These cutoff point is calculated as per the probability .
I will complete this table.
The value for first category will be 7.520 - (7.566-7.520) = 7.474
Expected Frequency for category 6 is 250/10 = 25, and the contribution of category 6 to the chi- square test statistic is (26-25)2 /25 = 0.04.
The combinded contribution of all the categories besides category 6 is 11.56. The chi- square test statistic is therfore equals to (11.56 + 0.04) = 11.6 and its P - value is 0.2368 .
Use a significance level of alpha = 0.01 to test the null hypothesis that the population of weights of Belgian 1 coins has a normal distributoion with a mean of 7.520 grams and a standard deviation of 0.036 Gram. The critical value is 21.666, and the null hypothesis is accepted.
Category Cutoff pint Obs. Freq. 1 7.474 20 2 7.490 27 3 7.501 31 4 7.511 20 5 7.520 30 6 7.529 26 7 7.539 18 8 7.550 31 9 7.566 17 10 N/A 30Related Questions
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