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Situation:

Question 1:
Researchers want to find a way to increase the growth rate of corn so it can be harvested more often. To test the effectiveness of a new type of spray fertilizer, plants grown from 200 corn seeds were divided equally into two groups: a treatment group and a control group. The treatment group was frequently sprayed with the new type of fertilizer. The growth in centimeters of each corn plant was noted before and after the treatment.

The results showed that the mean change in the growth rate by the treatment group is 8 points more than that of the control group. To test whether the results could be explained by random chance, researchers created a table that summarizes the results of 1,000 re-randomizations of the data with differences of means rounded to the nearest 2 points. Consider the significance level to be set at [tex]$5 \%$[/tex], so results less than [tex]$5 \%$[/tex] can be considered statistically significant.

\begin{tabular}{|c|c|}
\hline
Treatment Group Mean - Control Group Mean & Frequency \\
\hline
-12 & 1 \\
\hline
-10 & 10 \\
\hline
-8 & 28 \\
\hline
-6 & 58 \\
\hline
-4 & 125 \\
\hline
-2 & 184 \\
\hline
0 & 201 \\
\hline
2 & 186 \\
\hline
4 & 114 \\
\hline
6 & 57 \\
\hline
8 & 26 \\
\hline
10 & 8 \\
\hline
12 & 2 \\
\hline
\end{tabular}

Part A:

Determine the probability of the treatment group's mean being greater than the control group's mean by 8 points or more. Then complete the statements.

The significance level is set at [tex]$5 \%$[/tex], and the probability of the result is __%. The result is [tex]$\square$[/tex] \%, which is [tex]$\square$[/tex] the significance level.



Answer :

GinnyAnswer
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.