Table of contents

1. Intro to Stats and Collecting Data55m
2. Describing Data with Tables and Graphs1h 55m
3. Describing Data Numerically1h 45m
4. Probability2h 16m
5. Binomial Distribution & Discrete Random Variables2h 33m
6. Normal Distribution and Continuous Random Variables1h 38m
7. Sampling Distributions & Confidence Intervals: Mean1h 53m
8. Sampling Distributions & Confidence Intervals: Proportion1h 12m
- Sampling Distribution of Sample Proportion
  29m
- Confidence Intervals for Population Proportion
  42m
9. Hypothesis Testing for One Sample2h 19m
10. Hypothesis Testing for Two Samples3h 22m
11. Correlation1h 6m
12. Regression1h 4m
13. Chi-Square Tests & Goodness of Fit1h 20m
14. ANOVA1h 0m

4. Probability

Counting

Statistics

4. Probability

Counting

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1:39 minutes

Problem 3.4.31

Textbook Question

31. Experiment A researcher is randomly selecting a treatment group of 10 human subjects from a group of 20 people taking part in an experiment. In how many different ways can the treatment group be selected?

Verified step by step guidance

Step 1: Recognize that this is a combination problem because the order in which the subjects are selected does not matter. The formula for combinations is given by:

C = \frac{n!}{r! (n - r)!}

, where

n

is the total number of items, and

r

is the number of items to choose.

Step 2: Identify the values of

n

and

r

. Here,

n = 20

(total number of people) and

r = 10

(number of people to select).

Step 3: Substitute the values of

n

and

r

into the combination formula:

C = \frac{20!}{10! (20 - 10)!}

Step 4: Simplify the denominator by calculating

(20 - 10)!

, which equals

10!

. The formula now becomes:

C = \frac{20!}{10! 10!}

Step 5: Cancel out the common terms in the numerator and denominator, and compute the remaining terms to find the number of ways the treatment group can be selected. This involves dividing the factorial of 20 by the product of two factorials of 10.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Combinatorics

Combinatorics is a branch of mathematics dealing with combinations and permutations of objects. It provides the tools to count the number of ways to select items from a larger set without regard to the order of selection. In this context, it helps determine how many different groups of 10 can be formed from a total of 20 subjects.

Binomial Coefficient

The binomial coefficient, often denoted as 'n choose k' or C(n, k), represents the number of ways to choose k elements from a set of n elements without considering the order. It is calculated using the formula C(n, k) = n! / (k!(n-k)!), where '!' denotes factorial. This concept is essential for solving the problem of selecting the treatment group.

Factorial

A factorial, denoted as n!, is the product of all positive integers up to n. It is a fundamental concept in combinatorics used to calculate permutations and combinations. In the context of this question, factorials are used in the binomial coefficient formula to compute the total number of ways to select the treatment group from the available subjects.

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