A supervisor finds the mean number of miles that the employees in a 9the x has a line on top)department live from work. He finds X=2.9 and s=3. Which statement must be true?
z376 is within 1 standard deviation of the mean.
z37 is between 1 and 2 standard deviations of the mean.
z37 is between 2 and 3 standard deviations of the mean.
z37 is more than 3 standard deviations of the mean.
If I had a clue what the z in the answers meant...
I am a little wondering how the mean can be 2.9, and the standard deviation be 3. That is somewhat like saying the mean exam score was 83, and the standard deviation was 494.
To determine which statement is true, we need to calculate the z-score of the value 3.7 based on the given mean (X) and standard deviation (s).
The formula to calculate the z-score is:
z = (X - μ) / σ
Where:
X = Value being scored (3.7)
μ = Mean (2.9)
σ = Standard deviation (3)
Substituting the values into the formula, we get:
z = (3.7 - 2.9) / 3
Simplifying the calculation:
z = 0.8 / 3
z = 0.2667
Now, let's check if any of the given statements are true based on the calculated z-score:
Statement 1: z376 is within 1 standard deviation of the mean.
z376 = 0.2667
This statement is not true since 0.2667 is less than 1 standard deviation from the mean.
Statement 2: z37 is between 1 and 2 standard deviations of the mean.
This statement is not relevant to the given z-score of 0.2667.
Statement 3: z37 is between 2 and 3 standard deviations of the mean.
This statement is not relevant to the given z-score of 0.2667.
Statement 4: z37 is more than 3 standard deviations of the mean.
This statement is not relevant to the given z-score of 0.2667.
Therefore, none of the given statements is true based on the calculated z-score.
To determine which statement must be true, we need to calculate the z-score for the value z37. The z-score measures how many standard deviations a particular value is from the mean.
The formula to calculate the z-score is:
z = (x - μ) / σ
Where:
- x is the value we want to find the z-score for (in this case, z37)
- μ is the mean of the data (given as X = 2.9)
- σ is the standard deviation of the data (given as s = 3)
Let's substitute the values into the formula and calculate the z-score for z37:
z37 = (37 - 2.9) / 3
z37 = 34.1 / 3
z37 ≈ 11.3667
Now, let's evaluate each statement based on the z-score calculated:
1. z376 is within 1 standard deviation of the mean.
z376 = 37 - 2.9 / 3 ≈ 11.3667
This statement is false because the z-score is greater than 1.
2. z37 is between 1 and 2 standard deviations of the mean.
Since the z-score is greater than 1, this statement is also false.
3. z37 is between 2 and 3 standard deviations of the mean.
Since the z-score is greater than 2, this statement is also false.
4. z37 is more than 3 standard deviations of the mean.
Since the z-score is less than 3, this statement is true.
Therefore, the statement "z37 is more than 3 standard deviations of the mean" must be true.