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Statistics

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Apr 3, 2024

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Question 1 (a) 1.The percentage of the underfilled bottles will remain the same no matter how standard deviation change. 2. When standard deviation increase, the volume will tend to be away from the mean(302 ml). When standard deviation decrease, the volume will tend to be close to the mean(302 ml). (b) ● Percentage of underfilled bottles when μ=302 and σ=2 ml: 15.865525 ● Percentage of underfilled bottles when μ=302 and σ=1ml: 2.275013 ● Percentage of underfilled bottles when μ=302 and σ=0.5ml: 0.003167 ● Decreasing σ on the percentage of underfilled bottles causes decreasing of the percentage of underfilled bottles. Question 2 (a) (b) ● Shape of the histogram in part (a): ● The shape of the histogram is skewed right and unimodal, which supports the claim of the company that the bottles are slightly overfilled.

(c) This Q-Q plot confirms the results from the histogram because it is skewed to the right. (d) Summary statistics: Column n Mean Variance Std. dev. Std. err. Median Range Min Max Q1 Q3 volume 300 302.86567 0.3923624 3 0.626388 4 0.036164 551 302.7 2.8 302.1 304.9 302.4 303.2 ● The value of the mean is greater than the value of the median which suggests the data is right skewed. ● As for the quartiles, they are not equally distanced. The distance between median and Q3 is greater than the distance between the median and Q1, which suggests the data is right skewed.

Question 3 (a) (b) The shape of this histogram is bimodal and normally distributed. As the histogram in question 2 is unimodal and extremely right skewed, this histogram is bimodal and not skewed. The spread of the histograms are different as the range of the parent distribution is 302-305 which is larger than the range of this histogram, 302.4 to 303.4. (c)

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The plot confirms my findings in part b, as the data plotted are closer to the reference line, the second set of data is normally distributed. The second Q-Q plot shows more variations and smaller range due to smaller sample size. (d) Column Mean Std. dev. Mean 302.87767 0.24005408 The mean of the sampling sample is slightly higher than the mean sample. However, the standard deviation of the sampling sample is much higher than the mean sample. Question4 (a)

(b) The histogram is fairly bell-shaped and unimodal. It is normally distributed and slightly right skewed. Comparing to the histogram from question 2, this histogram has a much smaller spread 302 to 305 and 302.6 to 303.2,and less right skewed. Comparing to question 3, this histogram has a slightly smaller spread 302.4 to 303.4 and 302.6 to 303.2, and less right skewed. (c)

Comparing to the Q-Q plot in question 3, this Q-Q plot appears to be more normally distributed as the dots are very closed to the reference line. Thus we can conclude that the mean volume of 50 boxes of cola is normally distributed and as the sample size increases, the distribution become more normally distributed. (d) Column Mean Std. dev. Mean 302.85247 0.1064492 Both the standard deviation and the mean value of question 3 are greater than the the results from question 4. The mean obtained in question 3 is slightly closer to the population mean, 302.86567. 3b

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4.c) Obtain A QQ-Plot and a boxplot for the 50 means. Add a title to the plot. Paste Both Plots Into Your report. Do they confirm that the sample means indicate a normal distribution? Explain. Compare the Q-Q plot and boxplot with the Q-Q plot and boxplot obtained in part (c) of Question 3. What do you conclude?

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