Question

Let \(x\) be a binomial random variable with \(n=20\) and \(p=.1\). a. Calculate \(P(x \leq 4)\) using the binomial formula. b. Calculate \(P(x \leq 4)\) using Table 1 in Appendix I. c. Use the following Excel output to calculate \(P(x \leq 4)\). Compare the results of parts a, b, and c. d. Calculate the mean and standard deviation of the random variable \(x\). e. Use the results of part d to calculate the intervals \(\mu \pm \sigma, \mu \pm 2 \sigma,\) and \(\mu \pm 3 \sigma .\) Find the probability that an observation will fall into each of these intervals. f. Are the results of part e consistent with Tchebysheff's Theorem? With the Empirical Rule? Why or why not? Excel output for Exercise 33: Binomial with \(n=20\) and \(p=.1\) $$ \begin{array}{|c|c|c|c|c|} \hline & \mathrm{A} & \mathrm{B} & \mathrm{C} & \mathrm{D} \\ \hline 1 & \mathrm{x} & \mathrm{p}(\mathrm{x}) & \mathrm{x} & \mathrm{p}(\mathrm{x}) \\ \hline 2 & 0 & 0.1216 & 11 & 7 \mathrm{E}-07 \\ \hline 3 & 1 & 0.2702 & 12 & 5 \mathrm{E}-08 \\ \hline 4 & 2 & 0.2852 & 13 & 4 \mathrm{E}-09 \\ \hline 5 & 3 & 0.1901 & 14 & 2 \mathrm{E}-10 \\ \hline 6 & 4 & 0.0898 & 15 & 9 \mathrm{E}-12 \\ \hline 7 & 5 & 0.0319 & 16 & 3 \mathrm{E}-13 \\ \hline 8 & 6 & 0.0089 & 17 & 8 \mathrm{E}-15 \\ \hline 9 & 7 & 0.0020 & 18 & 2 \mathrm{E}-16 \\ \hline 10 & 8 & 0.0004 & 19 & 2 \mathrm{E}-18 \\ \hline 11 & 9 & 0.0001 & 20 & 1 \mathrm{E}-20 \\ \hline 12 & 10 & 0.0000 & & \\ \hline \end{array} $$

Short answer

Short Answer: We calculated the probability of \(P(x \leq 4)\) using the binomial formula, resulting in \(0.9578\), which was closely matched by the Excel output calculation of \(0.957\). The mean and standard deviation of the random variable were found to be \(\mu = 2\) and \(\sigma \approx 1.345\). The calculated probabilities for the intervals \(\mu \pm \sigma\), \(\mu \pm 2\sigma\), and \(\mu \pm 3\sigma\) were approximately \(0.7455\), \(0.957\), and \(0.9997\), respectively, and they were consistent with Tchebysheff's Theorem. The Empirical Rule wasn't directly applicable to the exercise, but the calculated probabilities were roughly similar to its estimates.

Step-by-step solution

(Step 1: Calculate \(P(x \leq 4)\) using the binomial formula)

To compute the probability of \(P(x \leq 4)\), we can sum the probabilities for each value of \(x\) up to and including \(4\). We'll use the binomial probability formula: $$P(x) = \binom{n}{x} p^x (1-p)^{(n-x)}$$ Now let's compute the probabilities for \(x \in \{0,1,2,3,4\}\) and sum them up: $$P(x \leq 4) = \sum_{x=0}^4 P(x) = \sum_{x=0}^4 \binom{20}{x} (0.1)^x (0.9)^{(20-x)}$$ After calculating the probabilities and summing them up, we get: $$P(x \leq 4) \approx 0.9578$$

(Step 2: Calculate \(P(x \leq 4)\) using Table 1 in Appendix I)

Unfortunately, we cannot specifically address this step as the required table (Table 1 in Appendix I) has not been provided with the exercise. You will have to refer to your textbook or source material for the table and locate the value of \(P(x \leq 4)\) for a binomial distribution with parameters \(n = 20\) and \(p = 0.1\).

(Step 3: Calculate \(P(x \leq 4)\) using the given Excel output)

Using the provided Excel output, we can find the probability \(P(x)\) for each value from \(0\) to \(4\), and then sum them up to get \(P(x \leq 4)\): $$P(x \leq 4) = P(0) + P(1) + P(2) + P(3) + P(4)$$ From the Excel output: $$P(x \leq 4) \approx 0.1216 + 0.2702 + 0.2852 + 0.1901 + 0.0898$$ $$P(x \leq 4) \approx 0.957$$ Comparing the results from parts a, b (not calculated here), and c, they should be very close or identical. Minor differences might occur due to rounding errors.

(Step 4: Calculate the mean and standard deviation of \(x\))

For a binomial random variable with parameters \(n\) and \(p\), the mean (\(\mu\)) and standard deviation (\(\sigma\)) can be calculated as follows: $$\mu = np$$ $$\sigma = \sqrt{np(1-p)}$$ Substituting the values: $$\mu = 20 \times 0.1 = 2$$ $$\sigma = \sqrt{20 \times 0.1 \times (1-0.1)} \approx 1.345$$ The mean is \(\mu = 2\) and the standard deviation is \(\sigma \approx 1.345\).

(Step 5: Calculate the intervals and probabilities for \(\mu \pm \sigma, \mu \pm 2\sigma\), and \(\mu \pm 3\sigma\))

We will now calculate the intervals and probabilities for \(\mu \pm \sigma, \mu \pm 2\sigma\), and \(\mu \pm 3\sigma\): 1. Interval \(\mu \pm \sigma\): \((2 - 1.345, 2 + 1.345) \approx (0.655, 3.345)\). Since \(x\) is a discrete variable, we will check the probabilities for \(x=1, 2, 3\): $$P(0.655 \leq x \leq 3.345) \approx P(1) + P(2) + P(3) = 0.2702 + 0.2852 + 0.1901 \approx 0.7455$$ 2. Interval \(\mu \pm 2\sigma\): \((2 - 2 \times 1.345, 2 + 2 \times 1.345) \approx (-0.69, 4.69)\). Check the probabilities for \(x=0, 1, 2, 3, 4\): $$P(-0.69 \leq x \leq 4.69) \approx P(0) + P(1) + P(2) + P(3) + P(4) \approx 0.957$$ 3. Interval \(\mu \pm 3\sigma\): \((2 - 3 \times 1.345, 2 + 3 \times 1.345) \approx (-2.035, 6.035)\). Check the probabilities for \(x=0, 1, 2, 3, 4, 5, 6\): $$P(-2.035 \leq x \leq 6.035) \approx P(0) + P(1) + P(2) + P(3) + P(4) + P(5) + P(6) \approx 0.9997$$

(Step 6: Check consistency with Tchebysheff's Theorem and the Empirical Rule)

Tchebysheff's Theorem states that for any random variable with mean \(\mu\) and standard deviation \(\sigma\), the proportion of the distribution lying within \(k\sigma\) of the mean is at least \(1 - \frac{1}{k^2}\). For \(k = 1\), Tchebysheff's Theorem gives a lower bound of \(1 - \frac{1}{1^2} = 0\). Our calculated probability for \(\mu \pm \sigma\) is \(0.7455\), which is consistent with Tchebysheff's Theorem. For \(k = 2\), Tchebysheff's Theorem gives a lower bound of \(1 - \frac{1}{2^2} = 0.75\). Our calculated probability for \(\mu \pm 2\sigma\) is \(0.957\), which is also consistent with Tchebysheff's Theorem. For \(k = 3\), the theorem gives a lower bound of \(1 - \frac{1}{3^2} = 0.889\). Our calculated probability for \(\mu \pm 3\sigma\) is \(0.9997\), which is consistent with Tchebysheff's Theorem. The Empirical Rule applies to normal distributions rather than binomial distributions, so it's not directly applicable to this exercise. However, it also provides probability estimates for intervals of \(\mu \pm \sigma, \mu \pm 2\sigma\), and \(\mu \pm 3\sigma\). For a normal distribution, the Empirical Rule states that approximately \(68\%\) of the data falls within \(\mu \pm \sigma\), \(95\%\) within \(\mu \pm 2\sigma\), and \(99.7\%\) within \(\mu \pm 3\sigma\). In our exercise, the calculated probabilities of \(0.7455, 0.957\), and \(0.9997\) are roughly similar to those given by the Empirical Rule, though it isn't directly applicable in this case.

Key concepts

Understanding Binomial Random Variables

A binomial random variable is the number of successes in a fixed number of independent trials. Each trial, often called a Bernoulli trial, has two outcomes: success or failure, and the probability of success in each trial is the same. For instance, if you flip a coin 20 times, and you're interested in the number of heads (successes) you get, with each coin flip being independent of the others, the number of heads is a binomial random variable.

Key characteristics of a binomial random variable include the number of trials (denoted as n), and the probability of success in a single trial (p). We denote the binomial random variable as X, which takes on values from 0 up to n, inclusive. Understanding the foundations of a binomial random variable is critical for effectively navigating more complex statistical theorems and rules. The exercise mentioned deals with n = 20 and p = 0.1, focusing on calculating specific probabilities associated with this distribution.

The Binomial Formula Explained

The binomial formula is a cornerstone for finding probabilities in a binomial distribution. Given the number of trials (n) and the probability of success (p), the probability of observing exactly x successes can be calculated using the formula:\[P(X = x) = \binom{n}{x} p^x (1-p)^{n-x}\]. It combines the concepts of combinations (denoted as \(\binom{n}{x}\), the number of ways to choose x successes from n trials) with the probabilities of successes and failures raised to the corresponding power. In the provided exercise, calculating \(P(X \leq 4)\) involves summing the probabilities from \(X = 0\) to \(X = 4\), with each calculated using the binomial formula.

Understanding the use of this formula is crucial as it lays the foundation for probability calculations across various scenarios in statistics.

Importance of Tchebysheff's Theorem

Tchebysheff's Theorem provides a very useful inequality that applies to all probability distributions with a defined mean (\(\mu\)) and variance (or standard deviation (\(\sigma\))). The theorem states that the probability of a random variable falling within k standard deviations of the mean is at least \(1 - \frac{1}{k^2}\) for k greater than 1. This is a powerful tool because it doesn't assume any particular shape of the distribution, so it's applicable even when the distribution is not normal. In our textbook exercise, Tchebysheff's Theorem ensures that even for a binomial distribution, which is typically not symmetric like the normal distribution, a certain proportion of the data will be found within a specified range around the mean.

Applying the Empirical Rule

While Tchebysheff's Theorem is universally applicable, the Empirical Rule is specifically designed for normal distributions and states that approximately 68%, 95%, and 99.7% of the data fall within one, two, and three standard deviations from the mean, respectively. Practically speaking, for data elements in a normal distribution, nearly all will fall within three standard deviations of the mean. Although the Empirical Rule doesn't strictly apply to binomial distributions, it provides a rule of thumb for approximation.

In our exercise with a binomial distribution, we compare the probabilities of intervals around the mean (\(\mu \pm \sigma\), \(\mu \pm 2\sigma\), \(\mu \pm 3\sigma\)) to the probabilities expected under normal distribution, as described by the Empirical Rule. This helps students understand the differences and limitations when applying such statistical rules.