1. A test has a mean of 50, standard deviation of 10.
A. What scores separate the middle 25% from the rest
b. What proportion would be expected to score between 45 and 55
c. What proportion would be expected to score 68 and above
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions and their related Z scores.
A. Find the Z score with .125 between score and mean. Insert ±Z value in the above equation and solve for the scores.
B and C. Solve for Z scores and use the table.
To answer these questions, we will use the concept of standard deviation and the Normal Distribution.
The Normal Distribution is a bell-shaped curve that describes the distribution of data when the mean and standard deviation are known.
A. To find the scores that separate the middle 25% from the rest, we need to find the z-scores associated with those percentiles. Z-scores are a measure of how many standard deviations a data point is away from the mean.
To find the z-scores, we can use the standard Normal Distribution table or calculate them using a formula. In this case, we will use the formula:
z = (x - μ) / σ
where z is the z-score, x is the data point, μ is the mean, and σ is the standard deviation.
Since we want the middle 25%, we need to find the z-scores for the lower and upper quartiles (Q1 and Q3). The middle 25% is equivalent to the interquartile range (IQR).
Q1 (lower quartile) z-score:
To find the z-score corresponding to the lower quartile, we use the formula:
z = (x - μ) / σ
And rearrange it to solve for x:
x = z * σ + μ
For the lower quartile, z = -0.674, as the lower quartile corresponds to the 25th percentile in the standard Normal Distribution table.
x = -0.674 * 10 + 50
x ≈ 43.26
So, the score that separates the lower 25% from the rest is approximately 43.26.
Q3 (upper quartile) z-score:
For the upper quartile, z = 0.674, as the upper quartile corresponds to the 75th percentile in the standard Normal Distribution table.
x = 0.674 * 10 + 50
x ≈ 56.74
So, the score that separates the upper 25% from the rest is approximately 56.74.
B. To find the proportion of scores expected to be between 45 and 55, we can use the z-scores for these data points.
z1 = (45 - 50) / 10
z2 = (55 - 50) / 10
Using the standard Normal Distribution table or a calculator with the Normal Distribution function, we can find the proportions associated with these z-scores.
The proportion between these two z-scores represents the proportion of scores between 45 and 55.
C. To find the proportion of scores expected to be 68 and above, we need to calculate the z-score for the score 68.
z = (68 - 50) / 10
Using the standard Normal Distribution table or a calculator with the Normal Distribution function, we can find the proportion associated with this z-score.