Sample Questions On Chart Reading, Probability, And Binomial

Sample Questionschart Reading Probability And Binomial Distribution

Read the provided data and questions related to probability, binomial distribution, and statistical analysis of daily customer counts, shoplifters, red lights run, and other activities over specified periods. Show all working out and use appropriate distribution models to analyze the data and answer the questions accurately.

Paper For Above instruction

The analysis of daily counts such as accidents, shoplifters, red light violations, customer numbers, and other activities requires a comprehensive understanding of probability, distribution models, and statistical inference. This paper systematically addresses each of the presented scenarios, employing relevant probability distributions—primarily binomial, Poisson, and normal distributions—and illustrates calculations step-by-step, justified by statistical reasoning.

Question 1: Accidents at a Dangerous Intersection

Given the recorded data of accidents per day over a year, the first step involves calculating the probability of no accidents occurring on a day. If the data shows the number of days with zero accidents, dividing that by total days provides the probability:

Let’s denote the number of days with zero accidents as \( d_0 \), and total days as 365. The probability \( P(\text{no accident}) \) is \( d_0/365 \).

Assuming the data shows, for example, 50 days with zero accidents, then:

\( P(\text{no accident}) = 50/365 \approx 0.137 \)

Similarly, the expected number of accidents per day is calculated using the mean from the data. If \( x_i \) represents the number of accidents on day \( i \), then:

\( \text{Expected accidents} = \frac{\sum_{i=1}^{365} x_i}{365} \)

For variance and standard deviation, the population standard deviation of accidents per day is computed using:

\( \sigma = \sqrt{\frac{\sum_{i=1}^{365} (x_i - \mu)^2}{365}} \)

Regarding the distribution, the Central Limit Theorem suggests that if the number of accidents is sufficiently large and independent, the distribution approximates normal. However, for small counts or skewed data, a Poisson distribution might be more suitable.

To analyze the number of days with no accidents over two weeks (14 days), the Binomial distribution is appropriate, considering each day as a Bernoulli trial with success probability \( p \) (probability of no accidents), and the number of successes (no accident days) as the binomial variable.

Question 2: Shoplifters Caught in a Store

Applying similar logic, the probability of a day with no shoplifter is obtained from the data, dividing days with zero shoplifters by total days. The expected number of shoplifters caught per day is the mean from the data, calculated as:

\( \lambda = \frac{\sum_{i=1}^{n} (\text{number of shoplifters on day } i)}{n} \)

The population standard deviation follows from the variance calculation across all days. Since shoplifters are discrete events that often occur rarely and independently, the Poisson distribution models the count of shoplifters per day effectively. For days with more than one shoplifter, Poisson is suitable; for small probabilities, binomial distribution could also be considered.

The number of days with multiple shoplifters over a period can be examined using binomial or Poisson models, depending on data specifics. For studying the probability of days exceeding one shoplifter over two weeks, the Poisson or negative binomial distribution can be utilized, supporting the assumption of rare and independent events.

Question 3: Red Lights Run at a Busy Intersection

Calculating the probability of no red lights run involves similar steps: count days with zero red lights and divide by total days. The expected number per day is the mean, and the standard deviation derived from the data informs about variability. The distribution choice—normal or Poisson—depends on counts; low counts are better modeled with Poisson, while higher counts may approximate normal.

The use of Poisson is supported by the data structure, where the counts are discrete and events are independent. For days with zero red lights,/or analyzing the distribution of counts, Poisson provides an appropriate model due to its suitability for count data of rare events.

Question 4 to 6: Customer Counts in a Restaurant

Customer numbers recorded over a year are analyzed to find the number of days within specific intervals, expected values, modes, probabilities, and distribution types. The proportion of days with customer counts between 110 and 130, for example, is estimated by dividing days in that range by total days.

The expected number of customers per day is derived from the mean. The mode of the histogram indicates the most frequent customer count-range, providing insights into typical customer volume. Estimations of probability for counts exceeding a threshold employ empirical data or theoretical distributions, commonly Poisson or normal, depending on data dispersion.

Calculations for probabilities such as “more than 90 customers” utilize the fitted distribution’s cumulative density function (CDF). The probability of at least one day in a three-day period with counts above 90 is obtained via binomial or Poisson models, considering the fixed probability per day.

For modeling the number of days exceeding 90 customers over the next 10 days, the binomial distribution is often suitable, assuming each day’s probability is independent and constant.

Questions 7 to 12: Application to Different Store Types and Time Series Analysis

The approach extends to similar data: calculating proportions, means, modes, and probability estimates. The key is selecting appropriate distribution models based on the data characteristics—Poisson for rare events, binomial for fixed number of trials, and normal for large counts with sufficient variance.

For time series data, trend analysis involves looking for patterns such as increasing, decreasing, or cyclic behaviors. Inferences about the future are based on the observed trend and possible seasonal effects, employing time series decomposition or regression models.

In all cases, the application of statistical inference, probability calculations, and distribution fitting enable comprehensive analysis of daily activity data, with subsequent decision-making or further research directed by these insights.

References

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  • Ross, S. M. (2014). Introduction to Probability Models (11th ed.). Academic Press.
  • Newbold, P., & Granger, C. W. J. (1974). Experiments with Econometric Models. Econometrica, 42(6), 1103–1126.
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