1. Suppose the amount of cola dispensed by a filling machine follows a normal distribution with a mean () and a standard deviation (). Select the Calculators option in the Stat menu and then the Normal option. This applet contains a graph of the normal density function and a calculator that enables you to calculate normal probabilities when the parameters ( and ) are provided . Use the applet to answer the following questions: (a) Assume that the mean amount dispensed by the machine is set at = 300 ml. Describe what happens to the percentage of underfilled bottles (the bottles containing less than 300 ml) when decreases or increases? In general, how does the magnitude of the standard deviation affect the filling process? (b) Now assume that the mean amount dispensed by the machine is set at = 302 ml. Enter the value of as 2 ml. Calculate the percentage of underfilled bottles (the bottles containing less than 300 ml) in this case. What is the percentage of underfilled bottles if were 1 ml and 0.5 ml? In general, what is the effect of decreasing on the percentage of underfilled bottles? 2. Consider a random sample of 300 bottles obtained from the population of all bottles filled by the machine over a specific short time period. The volume amount of cola in each bottle is determined. The 300 observations recorded in the column volume are available in the data file lab2a.txt in eClass. Given the very large sample size, we may assume that the distribution of the volume amount of cola in the sample is close enough to the population distribution and its mean and standard deviation are close to the population parameters ( and ). (a) Obtain a relative frequency histogram of the 300 observations with the bins starting at 302 and using a width of 0.5. Paste the histogram into your report. The format of the histogram should be the same as the format of the histogram in Lab 1 Instructions (labels at the axes, title). (b) Describe the shape of the histogram obtained in part (a). Does the histogram support the claim of the company that the bottles are slightly overfilled? Unimodal, skewed to the right, not symmetric. Yes, the mean of the 300 observations is 302.8ml. (c) Obtain the Q-Q plot for the 300 observations. Add a title to the plot. Paste the plot into your report. Does the plot confirm your findings in part (b) about the shape of the distribution? Yes because the majority of data is between 302 and 303 and relatively close to the line, the big line of data points just above the 302 mark, skews it to the right. (d) Use the Summary Statistics (Columns) feature to obtain the summary statistics (use the default options) for the 300 observations. Paste the summaries into your report. Is the relationship between the mean and median, as well as the relationship between the three quartiles, consistent with the observed shape of the histogram in part (b)? Column volume Summary statistics: n Mean Variance Std. dev. 300 302.86567 0.392362 43 0.626 3884 Std. err. Medi an Rang e 0.03616 4551 302.7 2.8 Min Max Q1 Q3 302.1 304.9 302.4 303.2 Since the histogram is skewed to the right that means the mean is larger than the median. The quartiles are also consistent with the histogram. Suppose that 50 boxes are randomly selected, each consisting of 6 bottles of cola obtained from the population of all bottles filled over a certain short time period. The amount of cola in each bottle is determined. The measurements are saved in a table consisting of 6 rows (sample size) and 50 columns (number of random samples) that occupies the columns Sample1(volume) - Sample50(volume) in the StatCrunch file lab2a.txt. 3. Obtain the mean amount of cola for each sample consisting of 6 bottles with the Summary Stats (Columns) feature and save the results in a column. Make sure that all 50 columns are included in the right panel of the Column Statistics dialog box. (a) Obtain a relative frequency histogram of the 50 means with the bins starting at 302.4 and using a width of 0.1. Paste the histogram into your report. The format of the histogram should be the same as the format of the histogram in Lab 1 Instructions (labels at the axes, title). (b) Refer to the histogram obtained in part (a). Do 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. It is not normally distributed, it is skewed to the right. The spread is larger than the histogram in question 2. It is skewed to a lesser degree than than question 2. (c) Obtain the Q-Q plot for the 50 means. Add a title to the plot. Paste the plot into your report. Does the plot confirm your findings in part (b)? Compare the plot with the one in part (c) of Question 2. Yes the plot confirms the findings in part(b). It is very largely spread compared to the plot in question 2. The data point are not as closely packed together and are farther away from the line. (d) Use the Summary Statistics (Columns) tool to obtain the mean, and standard deviation of the 50 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? Summary statistics: Column Mean of 50 samples Mean 302.87767 Std. dev. 0.24005408 Now suppose 50 boxes are randomly selected, each consisting of 30 bottles of cola obtained from the population of all bottles filled over the same short time period. The amount of cola in each bottle is determined and the measurements are saved in the StatCrunch file lab2b.txt in the form of a table of 50 columns, each consisting of 30 rows. 4. Obtain the mean amount of cola for each sample consisting of 30 observations with the Summary Stats (Columns) feature and save the results in a column. Make sure that all 50 columns are included in the right panel of the Column Statistics dialog box. (a) Obtain a relative histogram of the 50 means with the bins starting at 302.4 and using a width of 0.1. Paste the histogram into your report. The format of the histogram should be the same as the format of the histogram in Lab 1 Instructions (labels at the axes, title). (b) Describe the shape of the histogram in part (a). Do the data appear to be approximately normally distributed? Compare the histogram with the histogram obtained in part (a) of Question 2 and the one in part (a) of Question 3. In particular, comment about differences in spread and degree of skewness between the two distributions. (c) Obtain the Q-Q plot for the 50 means. Add a title to the plot. Paste the plot into your report. Does it look that the sample means come from a normal distribution? Explain. Compare the Q-Q plot with the Q-Q plot obtained in part (c) of Question 3. What do you conclude? (d) Use the Summary Statistics (Columns) feature to obtain the mean and standard deviation of the 50 means. Paste the summaries into your report. Compare the value of the standard deviation of the sample mean for n = 30 with the standard deviation of the sample mean in part (d) of Question 3 (for n = 6). 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. Which sample mean tends to be a more accurate estimate of the population mean? f933c6de22a1d6d6592088cd798c9e4892f52608 volume Sample1(volume) Sample2(volume) Sample3(volume) Sample4(volume) Sample5(volume) 303.6 302.4 303.9 302.8 303.9 302.1 302.2 303.6 303.3 303.7 304.1 302.6 302.3 302.1 302.5 304.1 302.5 302.1 302.7 302.4 302.7 302.9 302.1 302.5 302.3 302.6 303.2 302.3 303.4 304 303 303 302.1 303.5 302.6 303.2 302.1 302.2 302.8 302.1 302.4 304.1 302.2 303 303 302.4 303.1 304.2 303.6 303.3 303 302.8 302.6 304.6 302.4 302.8 302.2 302.8 302.4 302.6 302.4 302.2 302.1 303.1 303.9 302.1 303.9 302.1 302.1 303.2 303.4 303.1 302.7 303.3 303.2 303.7 303 303.3 303.5 302.2 304 302.3 Page 1 f933c6de22a1d6d6592088cd798c9e4892f52608 303.2 302.1 302.4 303.2 303 304.3 302.1 302.7 303.9 304.3 302.3 302.4 303.4 303.8 302.5 304.1 303.2 303.3 302.2 302.4 302.9 302.2 302.4 302.6 303.7 302.6 302.8 302.9 303.1 303.5 302.1 302.8 302.7 302.5 304.1 303.9 303.3 302.1 302.8 302.5 302.9 302.4 302.2 302.6 302.3 303.1 302.1 302.1 302.7 302.6 302.7 304.7 304.3 Page 2 f933c6de22a1d6d6592088cd798c9e4892f52608 302.3 302.1 302.8 302.3 302.3 303.2 302.2 303 302.8 302.5 302.4 302.7 302.4 303.2 304.6 302.5 302.4 302.6 302.1 302.5 302.5 302.4 302.4 302.1 302.9 302.2 303 302.3 303.7 303.7 303.1 303 302.1 304 302.5 303.9 303.7 304.9 302.5 302.5 302.3 302.1 302.7 302.8 303.2 304.7 302.5 303.1 302.4 302.1 302.6 302.3 303.4 Page 3 f933c6de22a1d6d6592088cd798c9e4892f52608 302.1 303.5 302.6 302.3 302.4 302.2 302.8 303.8 302.4 304.1 303.7 303.6 302.3 303.5 302.9 303.5 302.8 302.3 302.4 302.5 302.4 303.3 303.2 302.2 303.8 302.1 302.5 303.4 303.1 302.5 302.8 302.9 303.2 302.9 302.2 303.2 302.6 302.1 302.5 302.8 302.8 303.8 302.6 302.6 302.2 302.9 302.5 303.3 302.4 302.4 302.5 302.7 302.2 Page 4 f933c6de22a1d6d6592088cd798c9e4892f52608 302.6 303.4 303.9 303.2 302.5 302.8 302.6 302.7 302.1 302.6 303 303 302.3 302.5 303 302.8 302.4 303.8 302.3 303.2 303 303.3 302.8 302.6 303.8 302.1 303.7 302.9 303.6 303 303.2 302.6 302.6 303.3 303.9 302.3 302.9 303 303.5 302.6 303.1 302.3 302.5 302.1 303.4 302.2 302.9 303.2 302.7 303.2 303.2 302.6 302.2 Page 5 f933c6de22a1d6d6592088cd798c9e4892f52608 302.6 302.1 303.8 303.1 302.9 302.6 303.1 302.5 302.9 302.6 302.4 302.1 302.1 302.2 302.3 302.5 304.7 303.8 302.6 302.6 303.6 302.9 303.1 302.2 302.6 302.4 303.1 302.3 302.1 303.2 302.5 303.8 303.1 304.3 304.1 304.5 Page 6 f933c6de22a1d6d6592088cd798c9e4892f52608 Sample6(volume) Sample7(volume) Sample8(volume) Sample9(volume) Sample10(volume) 303 302.9 302.7 302.4 304.3 304.1 302.4 302.9 304 303.1 302.1 302.7 303.2 303.9 302.4 302.5 302.2 302.6 303.2 302.2 303.8 302.4 303 303 302.4 303.1 303.7 302.1 303.8 303.1 Page 7 f933c6de22a1d6d6592088cd798c9e4892f52608 Page 8 f933c6de22a1d6d6592088cd798c9e4892f52608 Page 9 f933c6de22a1d6d6592088cd798c9e4892f52608 Page 10 f933c6de22a1d6d6592088cd798c9e4892f52608 Page 11 f933c6de22a1d6d6592088cd798c9e4892f52608 Page 12 f933c6de22a1d6d6592088cd798c9e4892f52608 Sample11(volume) Sample12(volume) Sample13(volume) Sample14(volume) Sample15(volume) 302.6 302.1 302.5 303 302.6 302.3 303.3 303.2 303.4 302.4 303.9 302.5 302.1 302.1 302.2 302.1 303.6 304 302.3 303.8 302.2 302.9 302.1 302.8 303.2 303 302.9 302.7 302.7 302.8 Page 13 f933c6de22a1d6d6592088cd798c9e4892f52608 Page 14 f933c6de22a1d6d6592088cd798c9e4892f52608 Page 15 f933c6de22a1d6d6592088cd798c9e4892f52608 Page 16 f933c6de22a1d6d6592088cd798c9e4892f52608 Page 17 f933c6de22a1d6d6592088cd798c9e4892f52608 Page 18 f933c6de22a1d6d6592088cd798c9e4892f52608 Sample16(volume) Sample17(volume) Sample18(volume) Sample19(volume) Sample20(volume) 302.2 302.1 303.8 304.6 303.2 302.9 303.4 302.3 302.6 302.7 302.1 302.6 302.3 303.5 303 302.3 303.1 304.1 302.5 302.1 303 302.9 302.1 303 303.6 302.5 302.4 303.3 302.1 302.8 Page 19 f933c6de22a1d6d6592088cd798c9e4892f52608 Page 20 f933c6de22a1d6d6592088cd798c9e4892f52608 Page 21 f933c6de22a1d6d6592088cd798c9e4892f52608 Page 22 f933c6de22a1d6d6592088cd798c9e4892f52608 Page 23 f933c6de22a1d6d6592088cd798c9e4892f52608 Page 24 f933c6de22a1d6d6592088cd798c9e4892f52608 Sample21(volume) Sample22(volume) Sample23(volume) Sample24(volume) Sample25(volume) 303.8 302.7 302.3 302.5 302.4 302.5 303.9 302.9 302.5 302.6 303.2 303.1 302.9 302.8 302.5 302.9 302.4 302.6 302.9 302.5 302.4 302.1 302.3 302.4 303.1 302.6 302.2 302.2 303.9 302.5 Page 25 f933c6de22a1d6d6592088cd798c9e4892f52608 Page 26 f933c6de22a1d6d6592088cd798c9e4892f52608 Page 27 f933c6de22a1d6d6592088cd798c9e4892f52608 Page 28 f933c6de22a1d6d6592088cd798c9e4892f52608 Page 29 f933c6de22a1d6d6592088cd798c9e4892f52608 Page 30 f933c6de22a1d6d6592088cd798c9e4892f52608 Sample26(volume) Sample27(volume) Sample28(volume) Sample29(volume) Sample30(volume) 303 302.4 303 304.3 302.6 302.1 303.5 302.1 302.9 302.3 304.3 303.1 302.8 303.3 302.2 304.9 302.5 303.9 302.1 302.6 302.4 302.2 303.9 302.8 302.6 302.5 302.4 303.1 303.9 302.2 Page 31 f933c6de22a1d6d6592088cd798c9e4892f52608 Page 32 f933c6de22a1d6d6592088cd798c9e4892f52608 Page 33 f933c6de22a1d6d6592088cd798c9e4892f52608 Page 34 f933c6de22a1d6d6592088cd798c9e4892f52608 Page 35 f933c6de22a1d6d6592088cd798c9e4892f52608 Page 36 f933c6de22a1d6d6592088cd798c9e4892f52608 Sample31(volume) Sample32(volume) Sample33(volume) Sample34(volume) Sample35(volume) 302.3 302.2 302.2 302.3 302.5 302.1 304.1 302.1 303.3 302.1 302.7 302.7 303.8 303.5 302.9 302.8 302.9 302.9 303.4 304.3 302.9 303.5 302.8 303.3 302.1 303.2 303.2 303.2 302.1 303.1 Page 37 f933c6de22a1d6d6592088cd798c9e4892f52608 Page 38 f933c6de22a1d6d6592088cd798c9e4892f52608 Page 39 f933c6de22a1d6d6592088cd798c9e4892f52608 Page 40 f933c6de22a1d6d6592088cd798c9e4892f52608 Page 41 f933c6de22a1d6d6592088cd798c9e4892f52608 Page 42 f933c6de22a1d6d6592088cd798c9e4892f52608 Sample36(volume) Sample37(volume) Sample38(volume) Sample39(volume) Sample40(volume) 303 302.2 302.6 302.4 302.9 304.5 303.4 303.1 303.3 302.6 303.2 302.4 302.1 302.8 303 302.4 302.8 302.1 303 302.3 302.4 303.1 302.5 303 302.1 302.3 302.3 302.5 302.8 302.6 Page 43 f933c6de22a1d6d6592088cd798c9e4892f52608 Page 44 f933c6de22a1d6d6592088cd798c9e4892f52608 Page 45 f933c6de22a1d6d6592088cd798c9e4892f52608 Page 46 f933c6de22a1d6d6592088cd798c9e4892f52608 Page 47 f933c6de22a1d6d6592088cd798c9e4892f52608 Page 48 f933c6de22a1d6d6592088cd798c9e4892f52608 Sample41(volume) Sample42(volume) Sample43(volume) Sample44(volume) Sample45(volume) 304.1 302.8 302.1 302.3 303.2 302.1 302.3 302.9 302.4 302.2 303 303.1 302.7 303.1 302.3 303.9 304.1 302.2 302.1 303.5 304.3 302.9 303.8 302.6 302.6 302.8 304.7 302.6 303.8 304.3 Page 49 f933c6de22a1d6d6592088cd798c9e4892f52608 Page 50 f933c6de22a1d6d6592088cd798c9e4892f52608 Page 51 f933c6de22a1d6d6592088cd798c9e4892f52608 Page 52 f933c6de22a1d6d6592088cd798c9e4892f52608 Page 53 f933c6de22a1d6d6592088cd798c9e4892f52608 Page 54 f933c6de22a1d6d6592088cd798c9e4892f52608 Sample46(volume) Sample47(volume) Sample48(volume) Sample49(volume) Sample50(volume) 302.4 302.5 302.8 302.4 302.7 303.5 303.3 302.7 302.5 303.2 302.1 303.8 303.4 302.6 302.4 303.8 304.6 302.8 302.1 303.2 302.6 302.2 302.6 303.2 302.8 304.1 303.8 302.5 303.3 303.1 Page 55 f933c6de22a1d6d6592088cd798c9e4892f52608 Page 56 f933c6de22a1d6d6592088cd798c9e4892f52608 Page 57 f933c6de22a1d6d6592088cd798c9e4892f52608 Page 58 f933c6de22a1d6d6592088cd798c9e4892f52608 Page 59 f933c6de22a1d6d6592088cd798c9e4892f52608 Page 60 f933c6de22a1d6d6592088cd798c9e4892f52608 Column Mean of the 50 samples Sample1(volume) 302.6833333333 Sample2(volume) 302.95 Sample3(volume) 303.2166666667 Sample4(volume) 303.1 Sample5(volume) 302.75 Sample6(volume) 303.1 Sample7(volume) 302.7166666667 Sample8(volume) 302.75 Sample9(volume) 303.3833333333 Sample10(volume) 302.9166666667 Sample11(volume) 302.6833333333 Sample12(volume) 302.8833333333 Sample13(volume) 302.7666666667 Sample14(volume) 302.7166666667 Sample15(volume) 302.8333333333 Sample16(volume) 302.5 Sample17(volume) 302.75 Sample18(volume) 302.9833333333 Sample19(volume) 303.05 Sample20(volume) 302.9 Sample21(volume) 302.9 Sample22(volume) 302.7333333333 Sample23(volume) 302.5333333333 Sample24(volume) 302.8333333333 Sample25(volume) 302.6 Sample26(volume) 303.2 Sample27(volume) 302.6833333333 Sample28(volume) 303.1333333333 Sample29(volume) 303.2166666667 Sample30(volume) 302.4166666667 Sample31(volume) 302.6666666667 Sample32(volume) 303.1 Sample33(volume) 302.8333333333 Sample34(volume) 302.9833333333 Sample35(volume) 302.8333333333 Sample36(volume) 302.9666666667 Sample37(volume) 302.7 Sample38(volume) 302.4833333333 Sample39(volume) 302.8833333333 Sample40(volume) 302.5833333333 Sample41(volume) 303.3666666667 Sample42(volume) 303.3166666667 Sample43(volume) 302.7166666667 Sample44(volume) 302.7166666667 Sample45(volume) 303.0166666667 Sample46(volume) 303.0833333333 Sample47(volume) 303.3666666667 Sample48(volume) 302.8 Sample49(volume) 302.6833333333 Sample50(volume) 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