Identify The Unusual Values Of X In Each Histogram

Identify the unusual values of x in each histogram

Analyze the histograms presented, which depict binomial distributions with the same probability of success, p, but different numbers of trials, n. For each histogram, determine the unusual values of the variable x based on its distribution. Use the provided options to select the appropriate x values considered unusual in each case and justify your answer based on probability principles.

Sample Paper For Above instruction

In the analysis of binomial distributions, identifying unusual values of the random variable x is essential to understanding the variability and behavior of the data. The histograms in question represent binomial distributions with a fixed probability of success, p, but varying the number of trials, n. In particular, histogram (a) and histogram (b) display different distributions, and our goal is to determine the values of x that are considered unusual within each case.

Unusual values in a binomial distribution are typically those for which the probability of observing such a value is less than 0.05, or the values that fall outside the typical range around the mean (e.g., more than 2 standard deviations away). To identify these, we examine the shape, spread, and skewness of each histogram, alongside calculations of the mean and standard deviation, to ascertain which x-values are statistically unlikely or improbable according to the distribution's properties.

For histogram (a), which depicts a binomial distribution with a certain n, the central tendency around the mean suggests that values near the mean are most common. Values significantly lower or higher than this central range may be deemed unusual. The options provided include values like x = 5, 6, 7, 8 for histogram (a), and the decision hinges upon whether these fall within the typical range based on the distribution's standard deviation or probability thresholds.

Similarly, for histogram (b), distinguished by a different n, the possible unusual x-values are evaluated using the same criteria. If the distribution is skewed or centered differently, the range of typical values shifts accordingly. Values such as x = 4 or values like x=5, 6, 7, 8, may or may not be considered unusual depending on the specific parameters of the distribution, including its mean and variance.

In conclusion, the identification process involves understanding the shape of the binomial distributions, calculating the mean (np) and standard deviation (√npq), and then determining which x-values lie beyond the usual bounds (e.g., outside the range of mean ± 2 standard deviations). Values falling outside these bounds are considered statistically unusual, indicating they are less likely to occur under the assumed binomial model.

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