/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 22 (a) construct a discrete probabi... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 22

(a) construct a discrete probability distribution for the random variable \(X\) [Hint: \(\left.P\left(x_{i}\right)=\frac{f_{i}}{N}\right]\), (b) draw a graph of the probability distribution, (c) compute and interpret the mean of the random variable \(X,\) and \((d)\) compute the standard deviation of the random variable \(X\). Ideal Number of Children What is the ideal number of children to have in a family? The following data represent the ideal number of children for a random sample of 900 adult Americans. $$ \begin{array}{cc} x \text { (number of children) } & \text { Frequency } \\ \hline 0 & 10 \\ \hline 1 & 30 \\ \hline 2 & 520 \\ \hline 3 & 250 \\ \hline 4 & 70 \\ \hline 5 & 17 \\ \hline 6 & 3 \\ \hline \end{array} $$

Short Answer

Expert verified
P(X): [0.011, 0.033, 0.578, 0.278, 0.078, 0.019, 0.003]. Mean: 2.03. Standard Deviation: 0.91.

Step by step solution

01

- Calculate the Probability Distribution

To construct the discrete probability distribution for the random variable \(X\), calculate the probability of each \(x_i\). Use the formula \[P(x_i) = \frac{f_i}{N}\] where \(f_i\) is the frequency of \(x_i\) and \(N\) is the total number of observations. For the given data:\[ P(0) = \frac{10}{900} = 0.011 \]\[ P(1) = \frac{30}{900} = 0.033 \]\[ P(2) = \frac{520}{900} = 0.578 \]\[ P(3) = \frac{250}{900} = 0.278 \]\[ P(4) = \frac{70}{900} = 0.078 \]\[ P(5) = \frac{17}{900} = 0.019 \]\[ P(6) = \frac{3}{900} = 0.003 \].
02

- Represent the Probability Distribution Graphically

Using the probabilities calculated, draw a bar graph with the number of children (\(X\)) on the x-axis and their corresponding probabilities on the y-axis. Each bar should represent one probability value.For example: Plot \(X=0\) at \(0.011\), \(X=1\) at \(0.033\) and so on. Ensure the heights of the bars correspond accurately to the probabilities of each \(X\).
03

- Compute the Mean of the Random Variable \(X\)

The mean (expected value) of the random variable \(X\) is calculated by \[ E(X) = \sum (x_i \cdot P(x_i)) \]Using the probabilities from Step 1:\[ E(X) = (0 \cdot 0.011) + (1 \cdot 0.033) + (2 \cdot 0.578) + (3 \cdot 0.278) + (4 \cdot 0.078) + (5 \cdot 0.019) + (6 \cdot 0.003) = 2.029 \]So, the mean number of ideal children for the sample is approximately 2.03.
04

- Compute the Standard Deviation of \(X\)

The standard deviation \(\sigma\) is calculated using \[ \sigma = \sqrt{\sum [(x_i - E(X))^2 \, P(x_i)]} \]We need to compute the variance first. Using the mean \(E(X) = 2.03\):\[ \sigma^2 = [(0 - 2.03)^2 \cdot 0.011] + [(1 - 2.03)^2 \cdot 0.033] + [(2 - 2.03)^2 \cdot 0.578] + [(3 - 2.03)^2 \cdot 0.278] + [(4 - 2.03)^2 \cdot 0.078] + [(5 - 2.03)^2 \cdot 0.019] + [(6 - 2.03)^2 \cdot 0.003] = 0.83 \]Then take the square root of the variance to get the standard deviation:\[ \sigma = \sqrt{0.83} \approx 0.91 \].

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Key Concepts

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

Random Variable
A random variable is a concept in probability and statistics that represents possible outcomes of a random phenomenon. It is essentially a quantitative variable whose possible values depend on the outcomes of a random event.
In this exercise, the random variable, denoted as \(X\), represents the ideal number of children a family desires. Random variables can be either discrete or continuous:
鈥� A discrete random variable has specific values. In our example, \(X\) can take integral values from 0 to 6.
鈥� A continuous random variable, on the other hand, can take any value within a range.
Understanding the random variable context is crucial for interpreting probabilities and calculating important measures such as mean and standard deviation.
Probability Calculation
Calculating probabilities for a random variable involves determining the likelihood of each specific outcome.
For a discrete random variable, we use the formula:
\[ P(x_i) = \frac{f_i}{N} \]
where \(P(x_i)\) represents the probability of outcome \(x_i\), \(f_i\) is the frequency of \(x_i\), and \(N\) is the total number of observations. Breaking down the probabilities for our data on the ideal number of children, we get probabilities for each value of \(X\) based on their frequencies.
For example:
鈥� \( P(2) = \frac{520}{900} = 0.578 \)
鈥� \( P(3) = \frac{250}{900} = 0.278 \)
This step helps us construct a probability distribution, a function that provides the probabilities of occurrence of different possible outcomes.
Mean and Standard Deviation
The mean (or expected value) and standard deviation are crucial statistical measures.

The mean of a discrete random variable is calculated as:
\[ E(X) = \sum (x_i \cdot P(x_i)) \]
It gives us a central value or the average outcome of the random variable. For our example, the mean number of ideal children is calculated to be approximately 2.03.

The standard deviation measures the variability or dispersion of the random variable, providing insight into the amount of variation or spread in the data.
It is calculated as:
\[ \sigma = \sqrt{\sum [(x_i - E(X))^2 \cdot P(x_i)]} \]
In our case, it results in a standard deviation of about 0.91, indicating how much the ideal number of children can vary from the mean.

These calculations are essential as they help us understand the distribution and variability of the data, aiding in making informed interpretations and decisions.

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Most popular questions from this chapter

Determine whether the random variable is discrete or continuous. In each case, state the possible values of the random variable. (a) The number of light bulbs that burn out in the next week in a room with 20 bulbs. (b) The time it takes to fly from New York City to Los Angeles. (c) The number of hits to a website in a day. (d) The amount of snow in Toronto during the winter.

The phone calls to a computer software help desk occur at the rate of 2.1 per minute between 3:00 p.m. and 4:00 p.m. Compute the probability that the number of calls between 3:10 p.m. and 3:15 p.m. is (a) exactly eight. Interpret the result. (b) fewer than eight. Interpret the result. (c) at least eight. Interpret the result.

A binomial probability experiment is conducted with the given parameters. Compute the probability of \(x\) successes in the \(n\) independent trials of the experiment. $$ n=12, p=0.35, x \leq 4 $$

In the Healthy Hand washing Survey conducted by Bradley Corporation, it was found that \(64 \%\) of adult Americans operate the flusher of toilets in public restrooms with their foot. Suppose a random sample of \(n=20\) adult Americans is obtained and the number \(x\) who flush public toilets with their foot is recorded. (a) Explain why this is a binomial experiment. (b) Find and interpret the probability that exactly 12 flush public toilets with their foot. (c) Find and interpret the probability that at least 16 flush public toilets with their foot. (d) Find and interpret the probability that between 9 and 11 , inclusive, flush public toilets with their foot. (e) Would it be unusual to find more than 17 who flush public toilets with their foot? Why?

Determine whether the random variable is discrete or continuous. In each case, state the possible values of the random variable. (a) The amount of rain in Seattle during April. (b) The number of fish caught during a fishing tournament. (c) The number of customers arriving at a bank between noon and 1: 00 P.M. (d) The time required to download a file from the Internet.
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