Assume that hybridization experiments are conducted with peas having the property that for offspring, there is a 0.75 probability that a pea has green pods. Assume that the offspring peas are randomly selected in groups of 40. Complete parts (a) through (c) below.
a. Find the mean and the standard deviation for the numbers of peas with green pods in the groups of 40.
The value of the mean is p = 30 peas.
(Type an integer or a decimal. Do not round.)
The value of the standard deviation is o = 2.7 peas.
(Round to one decimal place as needed.)
b. Use the range rule of thumb to find the values separating results that are significantly low or significantly high.
Values of peas or fewer are significantly low.
(Round to one decimal place as needed.)

To solve this problem, we'll break down the given data and calculate each part step by step.

### Part (a): Mean and Standard Deviation
We are dealing with a binomial distribution since there are two possible outcomes for each pea (green pods or not green pods).

Given:
- Probability of a pea having green pods, \( p = 0.75 \)
- Number of trials (peas in the group), \( n = 40 \)

The **mean** \( \mu \) for a binomial distribution is calculated as:

\[
\mu = n \times p = 40 \times 0.75 = 30
\]

So the mean \( \mu = 30 \), which is already given.

The **standard deviation** \( \sigma \) for a binomial distribution is calculated as:

\[
\sigma = \sqrt{n \times p \times (1 - p)} = \sqrt{40 \times 0.75 \times (1 - 0.75)} = \sqrt{40 \times 0.75 \times 0.25}
\]

\[
\sigma = \sqrt{40 \times 0.1875} = \sqrt{7.5} \approx 2.7
\]

So, the standard deviation \( \sigma = 2.7 \), which is given.

### Part (b): Range Rule of Thumb

The **range rule of thumb** estimates the significantly low and significantly high values as being within two standard deviations of the mean.

- **Significantly low values**: These are values that are below \( \mu - 2\sigma \).
- **Significantly high values**: These are values that are above \( \mu + 2\sigma \).

Let’s calculate:

\[
\text{Low threshold} = \mu - 2\sigma = 30 - 2 \times 2.7 = 30 - 5.4 = 24.6
\]

Rounding to one decimal place, the significantly low value is **24.6** peas or fewer.

\[
\text{High threshold} = \mu + 2\sigma = 30 + 2 \times 2.7 = 30 + 5.4 = 35.4
\]

Rounding to one decimal place, the significantly high value is **35.4** peas or more.

### Summary:
- The mean is \( 30 \) peas.
- The standard deviation is \( 2.7 \) peas.
- Values of **24.6** peas or fewer are considered significantly low.
- Values of **35.4** peas or more are considered significantly high.

Would you like any further details or have any questions?

Here are 5 related questions:
1. How does the range rule of thumb help in identifying outliers in a data set?
2. What happens to the standard deviation if the sample size increases to 80 peas?
3. Can you explain how the binomial distribution differs from a normal distribution?
4. What are the implications of a higher probability, say \( p = 0.9 \), on the mean and standard deviation?
5. How would the results change if the number of peas was reduced to 20?

**Tip:** The range rule of thumb is a quick method, but for more accurate results, you can use precise statistical methods like z-scores or confidence intervals.

Assume that hybridization experiments are conducted with peas having the property that for offspring, there is a 0.75 probability that a pea has green pods. Assume that the offspring peas are randomly selected in groups of 40. Complete parts (a) through (c) below.

 a. Find the mean and the standard deviation for the numbers of peas with green pods in the groups of 40.
 The value of the mean is p = 30 peas.
 The value of the standard deviation is o = 2.7 peas.

 b. Use the range rule of thumb to find the values separating results that are significantly low or significantly high.

Learn how to calculate the mean and standard deviation for the number of peas with green pods in groups of 40, given a 0.75 probability of green pods. This problem also explores the range rule of thumb to determine significantly low and high values. Suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation