Test 3 Math 224 Type I November 6, 2013

Test 3 Math224 Type I November 06 2013s

Determine the probability distribution for the number of stops at gas stations or rest areas during a journey from Greensboro to Washington D.C., including filling in a missing value in the provided table. Calculate the probability that a traveler makes at least two stops. Find the expected value and standard deviation of the number of stops. Identify all Bernoulli (or binomial) experiments among the given scenarios, including surveying people's support for healthcare reform, counting daily car accidents, rolling a die multiple times, surveying students’ majors, and inspecting smartphones for defects. For a survey indicating 61% support for same-sex marriage with 7 randomly selected individuals, compute the probability that exactly three support it, and the probability that at least one supports it. Determine the expected value and standard deviation of the support count. Assess the truth of a given statement under certain probabilistic assumptions. Match probabilities between different scenarios and identify the differing value in a set of examples. Lastly, analyze a normally distributed variable, identifying an appropriate real-world variable, calculating the z-score for a given value, and computing specific probabilities and percentiles, including the 50th percentile.

Paper For Above instruction

The statistical analysis of the number of stops at gas stations or rest areas during a trip from Greensboro to Washington D.C. exemplifies the application of probability distributions in modeling real-world scenarios. Assuming the distribution is discrete and the data are available in a table with a missing probability value, the missing entry can be deduced by ensuring that the sum of all probabilities equals one. Filling in this value is essential for completeness and accuracy of the probability distribution, which then enables the calculation of related probabilities and statistical measures.

Once the distribution is complete, the probability that a traveler makes at least two stops can be calculated by summing the probabilities of making two or more stops. This cumulative probability offers insight into typical trip behaviors, indicating how often drivers plan for multiple stops. The expected value (mean) and standard deviation of the number of stops provide measures of central tendency and variability, respectively, enabling a comprehensive understanding of trip patterns. The formulas for expected value and standard deviation in discrete distributions are fundamental, providing the basis for analysis and interpretation.

In the context of identifying Bernoulli or binomial experiments, several scenarios classify as such based on their structure—repeat trials with two possible outcomes, fixed probabilities, and independence. For example, surveying whether individuals support healthcare reform or counting the number of successes (supporters) among a fixed number of trials fits the binomial model. Conversely, experiments like counting daily accidents or inspecting smartphones with multiple defect categories are not binomial but still valuable in probability modeling.

The survey data indicating that 61% of Americans support same-sex marriage facilitates binomial calculations. Given a sample of seven Americans, the probability that exactly three support the cause is calculated using the binomial probability formula: P(X=3) = C(7,3) (0.61)^3 (0.39)^4. The probability that at least one supports it is complementary to the probability that none do: 1 - P(X=0). The expected number of supporters and the standard deviation inform us about the anticipated support level and its variability, aiding in social research analyses.

Assessing the validity of a stated hypothesis often involves probability assumptions. For instance, assuming independence and identical distribution, one might evaluate whether a specific statement about the data distribution holds true. Matching probabilities across different scenarios and identifying outliers or differing values among accepted examples are critical analytical skills in statistical inference, reinforcing understanding of distribution properties.

Analyzing a normally distributed variable—such as human body temperature—requires identifying the variable, calculating the z-score for a specific value (e.g., 96.6°F), and using standard normal tables to determine associated probabilities. For example, the z-score is computed as (X - μ) / σ, where μ and σ are the mean and standard deviation, respectively. Probabilities for values below a certain point or within an interval are derived from the standard normal distribution, enabling predictions and percentile calculations. The median, or the 50th percentile, can be directly obtained from the symmetry of the normal distribution, matching the mean.

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