Mrs. Smith's reading class can read an average of 190 words per minute with a standard deviation of 20 words per minute. The top 3% of the class is to receive a special award. What is the minimum number of words per minute a student would need to read in order to get the award? Assume the data is normally distributed.
a. 228
b. 124
c. 226
d. 225
After four attempts on this one i kept getting different answers...can anyone lend a helping hand?
To find the minimum number of words per minute a student would need to read in order to get the award, we need to calculate the z-score corresponding to the top 3% of the class.
The z-score formula is given by:
z = (x - μ) / σ
where:
z = z-score
x = value of interest
μ = mean
σ = standard deviation
Since we are looking for the top 3%, we can find the z-score using the standard normal distribution table or calculator.
The z-score corresponding to the top 3% is approximately 1.88.
We can rearrange the z-score formula to solve for x:
x = z * σ + μ
Plugging in the values, we have:
x = 1.88 * 20 + 190
x = 37.6 + 190
x ≈ 227.6
Therefore, the minimum number of words per minute a student would need to read in order to get the award is approximately 227.6.
Among the options given, the closest value is 228, so the answer is (a) 228.
To find the minimum number of words per minute a student would need to read in order to get the award, we need to determine the cutoff point for the top 3% of the class.
Given that the data is normally distributed, we can use the Z-score formula to find the cutoff point:
Z = (X - μ) / σ
Where:
Z is the Z-score (represents the number of standard deviations from the mean)
X is the number of words per minute
μ is the mean (average) number of words per minute
σ is the standard deviation
In this case, the mean (μ) is given as 190 words per minute, and the standard deviation (σ) is given as 20 words per minute.
We want to find the Z-score that corresponds to the top 3% of the distribution. To do this, we can use a Z-table, or a calculator with a built-in Z-score function, to look up the Z-score that corresponds to a cumulative probability of 0.97 (1 - 0.03).
Using a Z-table or calculator, we find that the Z-score corresponding to a cumulative probability of 0.97 is approximately 1.88.
Now we can rearrange the Z-score formula to solve for X:
X = Z * σ + μ
Substituting in the known values:
X = 1.88 * 20 + 190
Calculating this, we get:
X ≈ 225.6
Rounding it up to the nearest whole number, the minimum number of words per minute a student would need to read in order to get the award is 226.
Therefore, the correct answer is c. 226.
what have you tried?
You can play around with Z table stuff at
http://davidmlane.com/hyperstat/z_table.html