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Sagot :
Certainly! Let's break down the problem and solve it step-by-step to find the probability and interpret the results using the given data.
### Step-by-Step Solution
1. Understand the Data:
Researchers conducted an experiment with 200 corn plants, divided into a treatment group and a control group. The mean growth rate difference (treatment group mean - control group mean) was assessed and summarized as the number of times each difference occurred across 1,000 re-randomizations.
2. Given Table:
The table provided shows the frequency of each observed difference in growth rates:
```
| Treatment Group Mean - Control Group Mean | Frequency |
|-------------------------------------------|-----------|
| -12 | 1 |
| -10 | 10 |
| -8 | 28 |
| -6 | 58 |
| -4 | 125 |
| -2 | 184 |
| 0 | 201 |
| 2 | 186 |
| 4 | 114 |
| 6 | 57 |
| 8 | 26 |
| 10 | 8 |
| 12 | 2 |
```
3. Total Number of Re-randomizations:
The total number of re-randomizations is 1,000.
4. Determine Relevant Frequencies:
We are interested in the instances where the treatment group's mean is greater than the control group's mean by 8 points or more. These relevant frequencies from the table are:
- For a difference of 8: 26
- For a difference of 10: 8
- For a difference of 12: 2
5. Sum of Relevant Frequencies:
Add these relevant frequencies:
[tex]\[ 26 + 8 + 2 = 36 \][/tex]
6. Calculate the Probability:
To find the probability, divide the relevant frequency sum by the total number of re-randomizations and then multiply by 100 to convert it to a percentage:
[tex]\[ \text{Probability} = \left(\frac{36}{1000}\right) \times 100 = 3.6\% \][/tex]
7. Interpret the Results:
The significance level is given as [tex]\(5\%\)[/tex]. We need to compare the calculated probability with this significance level to determine if the result is statistically significant.
### Final Statements Completion
Let’s fill in the blanks with the calculated probability and corresponding interpretation.
1. The significance level is set at [tex]\( 5\% \)[/tex], and the probability of the result is [tex]\(3.6\%\)[/tex].
2. Since [tex]\(3.6\%\)[/tex] is less than the [tex]\(5\%\)[/tex] significance level, we can conclude that the results are statistically significant.
### Completed Statements
"The significance level is set at [tex]$5\%$[/tex], and the probability of the result is [tex]$3.6\%$[/tex]. The result is [tex]$3.6\%$[/tex], which is less than the [tex]$5\%$[/tex] significance level."
Thus, based on the given data and calculations, the treatment group's mean being greater than the control group's mean by 8 points or more is statistically significant at the [tex]\(5\%\)[/tex] level.
### Step-by-Step Solution
1. Understand the Data:
Researchers conducted an experiment with 200 corn plants, divided into a treatment group and a control group. The mean growth rate difference (treatment group mean - control group mean) was assessed and summarized as the number of times each difference occurred across 1,000 re-randomizations.
2. Given Table:
The table provided shows the frequency of each observed difference in growth rates:
```
| Treatment Group Mean - Control Group Mean | Frequency |
|-------------------------------------------|-----------|
| -12 | 1 |
| -10 | 10 |
| -8 | 28 |
| -6 | 58 |
| -4 | 125 |
| -2 | 184 |
| 0 | 201 |
| 2 | 186 |
| 4 | 114 |
| 6 | 57 |
| 8 | 26 |
| 10 | 8 |
| 12 | 2 |
```
3. Total Number of Re-randomizations:
The total number of re-randomizations is 1,000.
4. Determine Relevant Frequencies:
We are interested in the instances where the treatment group's mean is greater than the control group's mean by 8 points or more. These relevant frequencies from the table are:
- For a difference of 8: 26
- For a difference of 10: 8
- For a difference of 12: 2
5. Sum of Relevant Frequencies:
Add these relevant frequencies:
[tex]\[ 26 + 8 + 2 = 36 \][/tex]
6. Calculate the Probability:
To find the probability, divide the relevant frequency sum by the total number of re-randomizations and then multiply by 100 to convert it to a percentage:
[tex]\[ \text{Probability} = \left(\frac{36}{1000}\right) \times 100 = 3.6\% \][/tex]
7. Interpret the Results:
The significance level is given as [tex]\(5\%\)[/tex]. We need to compare the calculated probability with this significance level to determine if the result is statistically significant.
### Final Statements Completion
Let’s fill in the blanks with the calculated probability and corresponding interpretation.
1. The significance level is set at [tex]\( 5\% \)[/tex], and the probability of the result is [tex]\(3.6\%\)[/tex].
2. Since [tex]\(3.6\%\)[/tex] is less than the [tex]\(5\%\)[/tex] significance level, we can conclude that the results are statistically significant.
### Completed Statements
"The significance level is set at [tex]$5\%$[/tex], and the probability of the result is [tex]$3.6\%$[/tex]. The result is [tex]$3.6\%$[/tex], which is less than the [tex]$5\%$[/tex] significance level."
Thus, based on the given data and calculations, the treatment group's mean being greater than the control group's mean by 8 points or more is statistically significant at the [tex]\(5\%\)[/tex] level.
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