Exercise A2 Use The Above Histogram To Determine The Probabi

Exercise A2use The Above Histogram Toa Determine The Probability T

Exercise A2: Use the above histogram to: (a) determine the probability that a random box of brand B raisins would fall between 25.8 and 29.4 (within one standard deviation of the mean). (b) calculate the values that are two standard deviations away from the mean. Sketch them on the graph. (c) determine the probability that a random raisin box would fall within two standard deviations of the mean. (d) Repeat steps b and c for raisin boxes that fall within three standard deviations from the mean. (e) Summarize your work from this exercise in a table like the one shown below.

Paper For Above instruction

The analysis of the histogram concerning the weights of boxes of brand B raisins offers valuable insights into the distribution and statistical measures of the data set. By examining these properties, we can better understand the probability of selecting a box within certain weight ranges and how these weights distribute around the mean. This essay systematically addresses each task outlined in the exercise, providing detailed steps and interpretations grounded in statistical theory.

Understanding the Histogram and Basic Statistical Measures

The first step involves interpreting the histogram to determine the mean (average) weight and the standard deviation, which measure the central tendency and dispersion of the data, respectively. The histogram visually displays the frequency of boxes falling within specific weight intervals. Extracting data points from the histogram enables calculating the mean by summing the products of each weight value and its corresponding frequency, then dividing by the total number of observations. Similarly, the standard deviation reflects how spread out the weights are around the mean, calculated using the squared deviations from the mean, averaged over the data set.

(a) Probability that a box weighs between 25.8 and 29.4 grams

Using the histogram, the probability that a randomly selected box weighs between 25.8 and 29.4 grams corresponds to the proportion of boxes within this interval. This involves summing the frequencies of all weight categories falling within this range and dividing by the total number of boxes sampled. The resultant probability indicates the likelihood that a randomly chosen box of brand B raisins falls within this weight interval, which is considered within one standard deviation from the mean assuming the distribution is approximately normal.

(b) Values two standard deviations away from the mean and sketching on the graph

Values two standard deviations away from the mean are calculated as follows: the lower bound is (mean - 2 × standard deviation), and the upper bound is (mean + 2 × standard deviation). These bounds encompass approximately 95% of the data in a normal distribution, according to the empirical rule. Graphically, these bounds are sketched on the histogram as vertical lines marking the limits. This visual representation aids in understanding the spread of the data and identifying the range covering most weights.

(c) Probability of a box falling within two standard deviations of the mean

The probability that a box’s weight is within two standard deviations from the mean can be determined by summing the frequencies of all categories falling inside the interval defined in (b). Dividing this sum by the total sample size yields the probability, which should approximate 0.95 if the data is normally distributed. This exercise confirms the empirical rule and aids in analyzing the distribution’s spread and shape.

(d) Repeat for three standard deviations from the mean

Extending this analysis, the bounds for three standard deviations are calculated as (mean - 3 × standard deviation) and (mean + 3 × standard deviation). These bounds typically encompass about 99.7% of the data, aligning with the empirical rule. By summing the frequencies within this broader interval and dividing by the total, we derive the probability of selecting a box within three standard deviations from the mean. Graphically, these bounds are added to the histogram, providing a comprehensive view of the data’s dispersion.

Summary and Tabular Presentation

Summarizing the calculations, the probabilistic measures at one, two, and three standard deviations are compiled into a table. This table includes the relevant ranges, corresponding probabilities, and their interpretation in the context of the distribution. Such a summary facilitates quick reference and comparison, allowing for an assessment of how well the data conforms to the expected properties of a normal distribution, and provides insights into the variability of the raisin box weights.

Conclusion

This analysis demonstrates the importance of statistical tools in understanding data distribution, especially in quality control contexts such as analyzing the weight consistency of food products. The histogram provides a visual representation, while the calculated probabilities quantify the likelihood of various weight ranges. Adherence to the empirical rule validates assumptions about the distribution, and the tabular summary consolidates key findings for clarity and decision-making.

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