School 1 Statistics: Average truancy per student is 8 days per semester with a standard deviation 2.2 and a population of 250 students.
a. What percent of the population is truant more than 9 days?
b. What percent of the population is truant less than 3 days?
c. What percent of the population is truant between 5 and 10 days?
d. What percent of the population is truant between 8.5 and 10 days?
e. What raw score would produce a truancy percent of the top 15% of the students who missed the least amount of school?
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions related to the various Z scores.
To answer these questions, we can use the concept of the standard normal distribution. This distribution is a probability distribution and it allows us to convert any normal distribution into a standard normal distribution with a mean of 0 and a standard deviation of 1.
To calculate probabilities using the standard normal distribution, we can use a Z-table or a statistical calculator. The Z-table provides the probability for a given raw score (Z-score). It tells us the percentage of data values that fall below a specific Z-score.
Now let's go through each question step by step:
a. To find the percentage of the population that is truant more than 9 days, we need to calculate the Z-score first. The formula for Z-score is:
Z-score = (x - μ) / σ
Where:
x = the raw score (9 days)
μ = the population mean (8 days)
σ = the population standard deviation (2.2 days)
Z-score = (9 - 8) / 2.2 = 0.45
Now, using the Z-table (or a calculator), we can find the area to the right of the Z-score 0.45. This represents the percentage of the population that is truant more than 9 days.
b. To find the percentage of the population that is truant less than 3 days, we can follow the same steps as in part a. Calculate the Z-score:
Z-score = (3 - 8) / 2.2 = -2.27
Now, we can use the Z-table to find the area to the left of the Z-score -2.27. This represents the percentage of the population that is truant less than 3 days.
c. To find the percentage of the population that is truant between 5 and 10 days, we can calculate two Z-scores:
For 5 days:
Z-score_1 = (5 - 8) / 2.2
For 10 days:
Z-score_2 = (10 - 8) / 2.2
Once we have the Z-scores, we can find the area between these two Z-scores using the Z-table. This will give us the percentage of the population that is truant between 5 and 10 days.
d. To find the percentage of the population that is truant between 8.5 and 10 days, we follow the same steps as in part c. Calculate the Z-score for 8.5 days and 10 days, and then find the area between these two Z-scores using the Z-table.
e. To find the raw score that would produce a truancy percent of the top 15% of the students who missed the least amount of school, we need to find the Z-score for the given percentile.
Using the Z-table, we can find the Z-score that corresponds to the top 15% of the distribution (the area to the left of this Z-score will be 0.15).
Once we have the Z-score, we can use the formula:
Z-score = (x - μ) / σ
Rearranging the formula, we can solve for x:
x = Z-score * σ + μ
Substitute the Z-score and the given values of μ and σ to find the raw score.
Note: The Z-table provides the area under the curve to the left of a given Z-score. If you need the area to the right of a Z-score, you can subtract the area from 1.
Using these steps, you can calculate the percentages and raw scores for each question.