3. A researcher wanted to study the effect of low light environments on the growth of alfalfa plants, so she grew six plants and measured their heights after two weeks. She calculated a sample mean of 5.2 cm with a standard deviation of 1.1 cm. If the heights of the first five plants in her sample were 4.1 cm, 4.9 cm, 3.9 cm, 6.0 cm, and 5.6 cm, what is the height of the other plant?

I keep getting 6.7 as the unknown length when I work backwards using the mean, but am not sure how the standard deviation works into the problem...

the s.d. is just a "smokescreen"

all you need is the mean and the heights

To calculate the height of the other plant, you can use the mean and standard deviation to find the z-score of the unknown plant and then convert it back to the original height using the formula for z-scores.

Here's how you can do it step by step:

1. Calculate the z-score:
The z-score measures how many standard deviations an individual value is from the mean. You can calculate the z-score using the formula:
z = (x - μ) / σ
Where:
- z is the z-score
- x is the value of the unknown plant's height
- μ is the mean height
- σ is the standard deviation

2. Substitute the known values:
In this case, the mean (μ) is given as 5.2 cm, and the standard deviation (σ) is given as 1.1 cm. So the formula becomes:
z = (x - 5.2) / 1.1

3. Calculate the z-score of the unknown plant:
To find the z-score, you can rearrange the equation and solve for x:
x = (z * 1.1) + 5.2

4. Calculate the z-score:
Now, you need to find the z-score for the unknown plant. Since it was not given in the problem, you can use the information provided about the heights of the other five plants in the sample. Calculate the mean of those five heights:
μ = (4.1 + 4.9 + 3.9 + 6.0 + 5.6) / 5 = 4.9 cm

5. Calculate the z-score of the unknown plant:
Now, you can calculate the z-score of the unknown plant by substituting the values into the formula from step 3:
x = (z * 1.1) + 4.9

6. Calculate the unknown height:
The problem asks for the height of the other plant, which corresponds to the unknown value (x). Use the calculated z-score from step 6 to find the unknown height:
x = (0 * 1.1) + 4.9
x = 4.9 cm

Therefore, the height of the other plant is 4.9 cm.