A score that is 10 points below the mean corresponds to a z-score of z=-2.00. What is the population standard deviation?
If z = -2, the score is 2 standard deviations below the mean. It is also 10 points below the mean.
-2*sigma = -10
The standard deviation (sigma) is therefore 5.
Well, if the mean is that much of a show-off that it's run off with 10 points, then it seems we have a z-score detective mystery on our hands! Let's call it "The Case of the Missing Standard Deviation."
To solve this puzzling enigma, we'll use the formula for z-score: z = (x - μ) / σ, where z is the z-score, x is the score we're interested in, μ is the mean, and σ is the standard deviation.
So, if we know the z-score (-2.00) and the mean, we can substitute those values into the formula and solve for the standard deviation: -2.00 = (x - μ) / σ. Since we know that x (the score) is 10 points below the mean, we can rewrite the equation as: -2.00 = (μ - 10 - μ) / σ, which simplifies to: -2.00 = -10 / σ.
Now, let's do some detective work and solve for σ. Multiplying both sides of the equation by σ, we get: -2.00σ = -10. Dividing both sides by -2.00 gives us: σ = 5.
That's right, ladies and gentlemen! After some sleuthing, we've discovered that the population standard deviation in this mysterious case is 5. Remember, this investigation may be over, but the clowning never stops!
To find the population standard deviation, we need to use the formula relating z-score to the mean and standard deviation:
z = (x - μ) / σ
where:
z is the z-score,
x is the data point,
μ is the population mean, and
σ is the population standard deviation.
From the given information, we have:
z = -2.00
x = μ - 10
Substituting these values into the formula, we have:
-2.00 = (μ - 10 - μ) / σ
Simplifying the equation, we get:
-2.00 = -10 / σ
To solve for σ (population standard deviation), we can cross-multiply:
-2.00 * σ = -10
Dividing both sides of the equation by -2.00, we find:
σ = -10 / -2.00
Simplifying further, we get:
σ = 5
Therefore, the population standard deviation is 5.
To find the population standard deviation, we need to use the z-score formula:
z = (x - μ) / σ
where:
- z is the z-score
- x is the raw score
- μ is the population mean
- σ is the population standard deviation
In this case, we are given that the z-score is -2.00, which corresponds to a score that is 10 points below the mean. Let's denote this score as x.
So, we have:
- z = -2.00
- x = μ - 10
Plugging in these values into the z-score formula, we get:
-2.00 = (μ - 10 - μ) / σ
Simplifying the equation, we have:
-2.00 = (-10) / σ
To isolate σ (the population standard deviation), we can cross-multiply:
-2.00 * σ = -10
Dividing both sides of the equation by -2.00, we find:
σ = -10 / -2.00
Calculating this expression, we have:
σ = 5
Therefore, the population standard deviation is 5.