2. A standardized exam was provided to all 3rd graders in Arizona schools. The average score was 75 with a standard deviation of 10. Assuming that the scores were normally distributed, answer the following questions. (a) What z-score corresponds with a score of 75? (b) What z-score corresponds with a score of 80? (c) What z-score corresponds with a score of 85? (d) What z-score corresponds with a score of 100? (e) What percent of students scored between a 75 and 85? (f) What percent of students scored between a 70 and 85? (g) What percent of students received a score of 95 or higher?
Here is the process.
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions/percentages related to the Z scores.
thanks for the answer
To answer these questions, we need to use the concept of z-scores. A z-score tells us how many standard deviations a particular value is away from the mean of a normal distribution.
(a) To find the z-score for a score of 75, we can use the formula:
z = (x - μ) / σ
where x is the given score, μ is the mean, and σ is the standard deviation.
Substituting the values, we get:
z = (75 - 75) / 10
z = 0
So, a score of 75 corresponds to a z-score of 0.
(b) To find the z-score for a score of 80, we can use the same formula:
z = (x - μ) / σ
Substituting the values, we get:
z = (80 - 75) / 10
z = 0.5
So, a score of 80 corresponds to a z-score of 0.5.
(c) Following the same calculation, a score of 85 corresponds to a z-score of 1.
(d) A score of 100 can be calculated as:
z = (100 - 75) / 10
z = 2.5
So, a score of 100 corresponds to a z-score of 2.5.
(e) To find the percent of students who scored between 75 and 85, we need to calculate the area under the normal curve between these two z-scores. This can be found by using a z-table or a statistical software. The z-scores for the given scores are 0 and 1.
(f) Similarly, to find the percent of students who scored between 70 and 85, we need to find the area under the normal curve between the z-scores of -0.5 and 1.
(g) To find the percent of students who received a score of 95 or higher, we need to find the area under the normal curve to the right of the z-score for a score of 95.
To calculate these areas, you can use a z-table or a statistical software that provides the cumulative probability values for the standard normal distribution.