91Ó°ÊÓ

In \(2006,\) the New York City Police Department stopped a half million pedestrians for suspected criminal involvement. Raw statistics for these encounters suggest large racial disparities \(-\) 89 percent of the stops involved nonwhites. Do these statistics point to racial bias in police officers' decisions to stop particular pedestrians? Do they indicate that officers are particularly intrusive when stopping nonwhites? 50\. Simulation According to the U.S. National Center for Health Statistics, there is a \(98 \%\) probability that a 20 -year-old male will survive to age \(30 .\) (a) Using statistical software, simulate taking 100 random samples of size 30 from this population. (b) Using the results of the simulation, compute the probability that exactly 29 of the 30 males survive to age 30 . (c) Compute the probability that exactly 29 of the 30 males survive to age \(30,\) using the binomial probability distribution. Compare the results with part (b). (d) Using the results of the simulation, compute the probability that at most 27 of the 30 males survive to age 30 . (e) Compute the probability that at most 27 of the 30 males survive to age 30 using the binomial probability distribution. Compare the results with part (d). (f) Compute the mean number of male survivors in the 100 simulations of the probability experiment. Is it close to the expected value? (g) Compute the standard deviation of the number of male survivors in the 100 simulations of the probability experiment. Compare the result to the theoretical standard deviation of the probability distribution. (h) Did the simulation yield any unusual results?

Step by step solution

01

Simulate 100 Random Samples

Using statistical software, simulate 100 random samples of size 30 from the population where each male has a 98% probability of surviving to age 30. Each simulation involves generating a binomial random variable with parameters n=30 and p=0.98.

02

Compute Probability of Exactly 29 Survivals from Simulation

Count the number of times exactly 29 out of 30 males survive in your 100 simulations. Divide this count by 100 to get the simulated probability.

03

Binomial Probability for Exactly 29 Survivals

Compute the probability using the binomial formula: \[ P(X = 29) = \binom{30}{29} (0.98)^{29} (0.02)^{1} \] Compare this result with the simulated probability from Step 2.

04

Probability of At Most 27 Survivals from Simulation

Count the number of times the number of survivors is 27 or fewer in your 100 simulations. Divide this count by 100 to get the simulated probability.

05

Binomial Probability for At Most 27 Survivals

Compute the probability using the binomial cumulative probability: \[ P(X \leq 27) = \sum_{k=0}^{27} \binom{30}{k} (0.98)^{k} (0.02)^{30-k} \] Compare this result with the simulated probability from Step 4.

06

Mean Number of Survivors

Compute the mean number of survivors in the 100 simulations: \[ \bar{X} = \frac{1}{100} \sum_{i=1}^{100} X_i \] where \( X_i \) is the number of survivors in each simulation. Compare this with the theoretical expectation \( E(X) = n \cdot p = 30 \cdot 0.98 \).

07

Standard Deviation

Calculate the standard deviation of the number of survivors in the 100 simulations: \[ s = \sqrt{\frac{\sum_{i=1}^{100} (X_i - \bar{X})^2}{99}} \] Compare this with the theoretical standard deviation \( \sigma = \sqrt{n \cdot p \cdot (1-p)} = \sqrt{30 \cdot 0.98 \cdot 0.02} \).

08

Evaluate Unusual Results

Review the frequency and distribution of observed results across the 100 simulations to determine if any unusual patterns or results were present when compared to theoretical expectations.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

binomial probability distribution

In the context of this exercise, we're dealing with a binomial probability distribution, which determines the probability of a specific number of successes in a fixed number of trials.
Here, a 'success' is defined as a 20-year-old male surviving to age 30. With each male having a 98% chance of survival, you're calculating the probabilities for different outcomes (like having exactly 29 out of 30 males survive).
To compute these probabilities, you use the binomial formula: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] where:

- \(P(X = k)\) is the probability of k successes in n trials
- \(\binom{n}{k}\) is the binomial coefficient, which can be calculated as \(\frac{n!}{k!(n-k)!}\)
- p is the probability of success on each trial (0.98 in this case)

For example, the probability that exactly 29 out of 30 males survive can be calculated by plugging in the values into the formula:
\[ P(X = 29) = \binom{30}{29} (0.98)^{29} (0.02)^{1} \]
This formula helps you understand how likely different outcomes are, based on the specified probability and sample size. It also allows for comparison with simulated results, providing a stronger grasp of practical versus theoretical probabilities.

statistical software

Using statistical software can make complex calculations and simulations much simpler and faster. In this exercise, statistical software helps in simulating 100 random samples of size 30 to understand the probabilities associated with survival rates.

Here's how statistical software enhances the process:

• **Easy Simulation**: It can quickly simulate multiple datasets according to specified conditions, like a 98% chance of survival.
• **Accurate Calculations**: The software accurately computes probabilities and other statistics, reducing human error.
• **Visualization**: It can generate visual representations (like graphs and histograms) for better conceptual understanding.

Specific steps like generating random samples, computing probabilities, and comparing results (as in parts b and c of the exercise) are straightforward with the appropriate software. For instance, using a program like R or Python, you can use built-in functions to generate binomial random variables and compute related probabilities. This not only saves time but also helps to focus on interpreting the results and understanding statistical concepts deeply.

standard deviation

Standard deviation is a measure of how spread out the values in a dataset are around the mean. In this exercise, calculating the standard deviation of the number of male survivors in 100 simulations helps understand the variability of the outcomes.

For a binomial distribution, the theoretical standard deviation is calculated using the formula: \[ \text{SD} = \sqrt{n \times p \times (1-p)} \] where:

• n is the number of trials (30 in this case)
• p is the probability of success (0.98)
• (1-p) is the probability of failure (0.02)

Plugging in the values, we get:
\[ \text{SD} = \sqrt{30 \times 0.98 \times 0.02} \]
Computing this gives an idea of how much the number of survivors might vary from the average (mean). In practice, you compare the computed standard deviation from your 100 simulations with the theoretical standard deviation to see if there are significant differences.

If your simulation values are close to the theoretical values, it indicates the model's accuracy in predicting real-world scenarios. Knowing this helps you understand the reliability of your statistical models and simulations.