For a particular sample of 63 scores on a psychology exam, the following results were obtained.
First quartile = 57 Third quartile = 87 Standard deviation = 9 Range = 51
Mean = 72 Median = 72 Mode = 98 Midrange = 57
Answer each of the following:
I. What score was earned by more students than any other score? Why?
II. What was the highest score earned on the exam?
III. What was the lowest score earned on the exam?
IV. According to Chebyshev's Theorem, how many students scored between 45 and 99?
V. Assume that the distribution is normal. Based on the Empirical Rule, how many students scored between 45 and 99?
Please show all of your work.
I-III
mode:http://www.mathgoodies.com/lessons/vol8/mode.html
midrange: http://en.wikipedia.org/wiki/Mid-range
IV. Chebyshev's Theorem: http://www.statisticshowto.com/articles/how-to-calculate-chebyshevs-theorem/
You know the value of mean and standard deviation. Find the number of standard deviations (Find x)!
72 +/- (x)(9) = (45, 99)
V. Empirical 68-95-99.7 Rule
http://www.stat.sfu.ca/~cschwarz/Stat-201/Handouts/node37.html
I. To determine the score earned by more students than any other score, we need to identify the mode of the data set. The mode is the value that appears most frequently in the data. In this case, the mode is given as 98, which means more students earned a score of 98 than any other score.
II. The highest score earned on the exam can be determined by finding the maximum value in the data set. However, it is not provided in the given information. Therefore, we cannot determine the highest score without additional data.
III. Similarly, the lowest score earned on the exam is not provided in the given information. Therefore, we cannot determine the lowest score without additional data.
IV. According to Chebyshev's Theorem, we can determine the proportion of scores within a certain number of standard deviations from the mean. Chebyshev's Theorem states that regardless of the shape of the distribution, at least (1 - 1/k^2) of the data falls within k standard deviations of the mean, where k is any value greater than 1.
In this case, to determine how many students scored between 45 and 99, we need to calculate the proportion of scores within two standard deviations from the mean. Since the standard deviation is given as 9, k = 2. By applying Chebyshev's Theorem, at least (1 - 1/2^2) = (1 - 1/4) = 3/4 = 75% of the data falls within two standard deviations from the mean. Therefore, at least 75% of the students scored between 45 and 99.
V. The Empirical Rule, also known as the 68-95-99.7 rule, is applicable when the distribution is approximately normal. It states that in a normal distribution:
- Approximately 68% of the data falls within one standard deviation of the mean.
- Approximately 95% of the data falls within two standard deviations of the mean.
- Approximately 99.7% of the data falls within three standard deviations of the mean.
Since the data is not explicitly stated to follow a normal distribution, we cannot directly apply the Empirical Rule. However, if we assume the distribution is normal, we can estimate the proportion of students who scored between 45 and 99.
To apply the Empirical Rule, we need to calculate the mean and standard deviation of the data, but these values are not provided. Therefore, we cannot determine the number of students who scored between 45 and 99 using the Empirical Rule without additional data.