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
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?
Confused!!
I can help you answer those questions step-by-step. Let's break them down:
I. What score was earned by more students than any other score? Why?
The mode, which is 98, represents the score earned by more students than any other score. This is because it has the highest frequency or occurrence in the data set.
II. What was the highest score earned on the exam?
The highest score earned on the exam is determined by the range of the data, which is 51. Since the range is the difference between the highest and lowest scores, we add the range to the lowest score to get the highest score. So, if the lowest score is X, then the highest score can be calculated as X + 51.
III. What was the lowest score earned on the exam?
The lowest score earned on the exam can be determined by subtracting the range from the highest score. So, if the highest score is Y, then the lowest score can be calculated as Y - 51.
IV. According to Chebyshev's Theorem, how many students scored between 45 and 99?
Chebyshev's Theorem states that for any data set, regardless of the shape of the distribution, at least (1 - 1/((k^2))) of the data falls within k standard deviations of the mean. In this case, we can calculate the number of students who scored between 45 and 99 by using Chebyshev's Theorem as follows:
Let's assume k as 3 (This is often used in practice).
So, at least (1 - 1/(3^2)) = (1 - 1/9) = 8/9 of the data falls within 3 standard deviations of the mean.
Using the formula, we can calculate the range within 3 standard deviations of the mean as follows:
Range = (3 × standard deviation)
Therefore, the range within 3 standard deviations of the mean is 3 × 9 = 27.
Since the mean is 72, the range between 45 and 99 falls within 3 standard deviations of the mean.
So, according to Chebyshev's Theorem, at least 8/9 of the students will score within 3 standard deviations of the mean.
V. Assume that the distribution is normal. Based on the Empirical Rule, how many students scored between 45 and 99?
The Empirical Rule is a rule of thumb that applies to normal distributions. According to the Empirical Rule:
- Approximately 68% of the data falls within 1 standard deviation of the mean.
- Approximately 95% of the data falls within 2 standard deviations of the mean.
- Approximately 99.7% of the data falls within 3 standard deviations of the mean.
Assuming the distribution is normal, we can calculate the approximate percentage of students who scored between 45 and 99:
- For 1 standard deviation: Approximately 68% of the students scored between 45 and 99.
- For 2 standard deviations: Approximately 95% of the students scored between 45 and 99.
- For 3 standard deviations: Approximately 99.7% of the students scored between 45 and 99.
Since we are assuming a normal distribution, we can use these approximations to estimate the number of students who scored between 45 and 99.
I'm here to help! Let's break down each question step by step:
I. What score was earned by more students than any other score? Why?
To determine the score earned by more students than any other score, we need to look at the mode. The mode is the value that appears most frequently in the data set. According to the information provided, the mode is 98. This means that more students earned a score of 98 than any other score.
II. What was the highest score earned on the exam?
To find the highest score earned on the exam, we look at the range. The range is the difference between the highest and lowest scores. From the given information, the range is 51. Therefore, the highest score must be the lowest score (in this case, we do not have further information to determine the exact highest score).
III. What was the lowest score earned on the exam?
Similarly, to find the lowest score earned on the exam, we again refer to the range. The range is the difference between the highest and lowest scores, which in this case is given as 51. So the lowest score is the highest score minus the range, which is not provided in the given information.
IV. According to Chebyshev's Theorem, how many students scored between 45 and 99?
Chebyshev's Theorem states that for any data set, regardless of its shape, at least (1-1/𝑘^2) of the data must be within 𝑘 standard deviations of the mean. In this case, the standard deviation is given as 9. To find the number of students who scored between 45 and 99, we need to calculate 𝑘.
𝑘 can be determined by finding the absolute difference between the mean and the upper or lower bound (99 or 45) and dividing it by the standard deviation.
For the upper bound (99):
𝑘 = |99 - 72| / 9 = 27 / 9 = 3
For the lower bound (45):
𝑘 = |45 - 72| / 9 = 27 / 9 = 3
Using 𝑘 = 3, we can substitute it into (1 - 1/𝑘^2) to get the proportion of data within 3 standard deviations of the mean. 1 - 1/3^2 = 1 - 1/9 = 8/9.
Therefore, according to Chebyshev's Theorem, at least 8/9 of the data (or 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?
The Empirical Rule is applicable to normally distributed data. It states that for a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, approximately 95% falls within two standard deviations, and approximately 99.7% falls within three standard deviations.
Since we already know from the previous question that the distribution has a standard deviation of 9, we can use the Empirical Rule.
Within one standard deviation of the mean (72 ± 9), approximately 68% of the students' scores would fall. This means that approximately 68% of the students scored between 63 and 81.
Within two standard deviations of the mean (72 ± 2 * 9), approximately 95% of the students' scores would fall. This means that approximately 95% of the students scored between 54 and 90.
Within three standard deviations of the mean (72 ± 3 * 9), approximately 99.7% of the students' scores would fall. This means that approximately 99.7% of the students scored between 45 and 99.
Therefore, assuming a normal distribution, approximately 99.7% of the students, or almost all of them, scored between 45 and 99.