The mean of Exam 1 is 80%. The variance is 25. How many students of my IS310 class of 200 students have scores between 70% and 90%, assuming that the distribution is bell-shaped?
a.180
b.185
c.190
d.195
I'm not sure how to set up this problem.
Z = (score-mean)/SEm
SEm = SD/√n
SD = √variance
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability for each and the Z score. Insert into above equation.
I'm not allowed to use the chart. I have to calculate it by hand
So I think the way the professor wanted to answer this is knowing the empirical rule because its normal
65%= 1 SD
95%=2 SD
98%=3 SD
all from the mean
Variance of 25 gives us a Standard deviation of 5.
80% is the mean. So moving 2 SD up gives us 90%
Its normal and the empirical rule states that 2 SD is 95%
So you multiply (95%)x 200= 190
To solve this problem, we need to use the concept of the standard deviation and the z-score. The z-score measures how many standard deviations a data point is away from the mean of a distribution.
First, let's find the standard deviation. The variance is given as 25, so the standard deviation (denoted as σ) is the square root of the variance, which is √25 = 5.
Next, we can find the z-scores for the lower and upper boundaries. To find the z-score, we can use the formula:
z = (x - μ) / σ
where x is the score, μ is the mean, and σ is the standard deviation.
For the lower bound, x = 70, μ = 80, and σ = 5. Plugging these values into the formula gives us:
z_lower = (70 - 80) / 5 = -10 / 5 = -2
For the upper bound, x = 90, μ = 80, and σ = 5. Plugging these values into the formula gives us:
z_upper = (90 - 80) / 5 = 10 / 5 = 2
Now, we need to find the area under the bell-shaped distribution curve between these two z-scores. This area represents the proportion of students with scores between 70% and 90%.
Using a z-table or a statistical calculator, we can find that the area between z = -2 and z = 2 is approximately 0.9545. This means that approximately 95.45% of the students have scores between 70% and 90%.
Finally, we can calculate the number of students by multiplying the proportion by the total number of students:
Number of students = Proportion * Total number of students
= 0.9545 * 200
≈ 190
Therefore, the answer is c. 190.