**Problem Overview:**Red and blue pens are sold in packets of six. Each pen is either red or blue. The manufacturer claims that the number of red pens in a packet is binomially distributed. He collects 100 packets at random and obtains the information in the table.| Number of red pens | Number of packets ||--------------------|-------------------|| 0 | 1 || 1 | 3 || 2 | 9 || 3 | 17 || 4 | 31 || 5 | 28 || 6 | 11 |**Questions:**a) State the null and the alternative hypothesis.b) Calculate the mean number of red pens per packet.c) Hence estimate the probability that a randomly chosen pen is red.---**Solution Explanation:**a) **State the null and the alternative hypothesis** - Null Hypothesis: The number of red pens in a packet follows a binomial distribution. - Alternative Hypothesis: The number of red pens in a packet does not follow a binomial distribution.b) **Calculate the mean number of red pens per packet** - To calculate the mean, multiply the number of red pens by the number of packets for each category, sum these products, and then divide by the total number of packets. \[ \text{Mean} = \frac{(0 \times 1) + (1 \times 3) + (2 \times 9) + (3 \times 17) + (4 \times 31) + (5 \times 28) + (6 \times 11)}{100} \] \[ \text{Mean} = \frac{0 + 3 + 18 + 51 + 124 + 140 + 66}{100} = \frac{402}{100} = 4.02 \]c) **Estimate the probability that a randomly chosen pen is red** - The probability of a pen being red in a packet is the mean number of red pens per packet divided by the total number of pens in a packet. \[ \text{Probability} = \frac{4.02}{6} \approx 0.67 \]Hence, there is an estimated probability of 0.

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Red and blue pens are sold in packets of sıx. Each pen is either red or blue. The manufacturer claims that the number of red pens in a packet is binomially distributed. He collects 100 packets at random and obtains the information in the table. Number of Number of packets 1 red pens 1 3 17 31 28 11

Red and blue pens are sold in packets of sıx. Each pen is either red or blue. The manufacturer claims that the number of red pens in a packet is binomially distributed. He collects 100 packets at random and obtains the information in the table. Number of Number of packets 1 red pens 1 3 17 31 28 11

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Concept explainers

Contingency Table

A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.

Binomial Distribution

Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.

Topic Video

Question

**Problem Overview:**

Red and blue pens are sold in packets of six. Each pen is either red or blue. The manufacturer claims that the number of red pens in a packet is binomially distributed. He collects 100 packets at random and obtains the information in the table.

| Number of red pens | Number of packets |
|--------------------|-------------------|
| 0 | 1 |
| 1 | 3 |
| 2 | 9 |
| 3 | 17 |
| 4 | 31 |
| 5 | 28 |
| 6 | 11 |

**Questions:**

a) State the null and the alternative hypothesis.

b) Calculate the mean number of red pens per packet.

c) Hence estimate the probability that a randomly chosen pen is red.

---

**Solution Explanation:**

a) **State the null and the alternative hypothesis**
- Null Hypothesis: The number of red pens in a packet follows a binomial distribution.
- Alternative Hypothesis: The number of red pens in a packet does not follow a binomial distribution.

b) **Calculate the mean number of red pens per packet**
- To calculate the mean, multiply the number of red pens by the number of packets for each category, sum these products, and then divide by the total number of packets.

\[
\text{Mean} = \frac{(0 \times 1) + (1 \times 3) + (2 \times 9) + (3 \times 17) + (4 \times 31) + (5 \times 28) + (6 \times 11)}{100}
\]

\[
\text{Mean} = \frac{0 + 3 + 18 + 51 + 124 + 140 + 66}{100} = \frac{402}{100} = 4.02
\]

c) **Estimate the probability that a randomly chosen pen is red**
- The probability of a pen being red in a packet is the mean number of red pens per packet divided by the total number of pens in a packet.

\[
\text{Probability} = \frac{4.02}{6} \approx 0.67
\]

Hence, there is an estimated probability of 0.

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