Question

For a sample of n = 30 scores, X = 45 corresponds to z = 1.50 and X = 40 corresponds to z = +1.00. What are the values for the sample mean and standard deviation?

Answer 1

The sample mean (μ) and standard deviation (σ) are calculated using the provided z-scores and corresponding sample values, resulting in a mean of 30 and a standard deviation of 10.

Step-by-step explanation:

To solve for the sample mean and standard deviation using the given z-scores and sample values, we can employ the formula for calculating a z-score:

z = (X - μ) / σ

where X is the sample value, μ is the mean, and σ is the standard deviation. For X = 45 with a z-score of 1.50, the equation is:

1.50 = (45 - μ) / σ

For X = 40 with a z-score of 1.00, the equation becomes:

1.00 = (40 - μ) / σ

By solving these two equations simultaneously, we can find the values of μ and σ.

From the first equation, we have:

1.50σ = 45 - μ

From the second equation, we have:

1.00σ = 40 - μ

If we multiply the second equation by 1.5, it becomes:

1.50σ = 60 - 1.5μ

We can set the expressions for 1.50σ equal to each other:

45 - μ = 60 - 1.5μ

Solving for μ gives us the sample mean:

μ = 60 - 45 = 15 / (1.5 - 1) = 15 / 0.5 = 30

To find the standard deviation σ, we substitute μ = 30 into one of the original equations:

1.50σ = 45 - 30 = 15

Therefore, σ = 15 / 1.50 = 10

So, the sample mean (μ) is 30 and the sample standard deviation (σ) is 10.

Answer 2

30

Step-by-step explanation:

Here, it is given that the Z – score corresponding to the score X = 45 is 1.5 and the Z – score corresponding to the score X = 40 is...

Z =
`(X - M)/(S)`

So:

1/ 1.5 =
`(45 - M)/(S)` and 1 =
`(40 -M)/(S)` . solve the both equations, we found out:

<=> S = 10

=> M = 30

Hence, mean is:

M = 45 -1.5S = 45 - 1.5 (10) = 30