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!!
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1. Mode
2. (max+min)/2 = 57 (midrange)
max+min = 114
but max-min = 51
Add the two equations.
2(max) = 165, max = 82.5?
3. Continue with 2?
4 & 5. 45 and 99 are mean ± 3 SD. What does that tell you?
No worries! I'll help you understand and find the answers to each of your questions step by step.
I. What score was earned by more students than any other score? Why?
To find the score earned by more students than any other score, we need to determine the mode. From the given information, we know that 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 need to determine the range. The range is the difference between the highest and lowest scores. From the given information, we know that the range is 51. To find the highest score, we add the range to the lowest score. So, the highest score earned on the exam is 57 + 51 = 108.
III. What was the lowest score earned on the exam?
From the given information, we know that the lowest score is 57.
IV. According to Chebyshev's Theorem, how many students scored between 45 and 99?
Chebyshev's Theorem provides a general inequality for any data set. According to Chebyshev's Theorem, at least (1 - 1/k^2) * 100% of the data lies within k standard deviations of the mean, where k is any positive number greater than 1.
In this case, the standard deviation is given as 9. Let's use k = 3 to calculate the minimum proportion of data lying between 45 and 99. (1 - 1/3^2) * 100% = 1 - 1/9 = 8/9 ≈ 0.8889
Therefore, at least 88.89% of the students will score between 45 and 99, according to Chebyshev's Theorem.
V. Assume that the distribution is normal. Based on the Empirical Rule, how many students scored between 45 and 99?
The Empirical Rule, also known as the 68-95-99.7 rule, applies only to data sets that have a normal distribution. It states that:
- 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.
Given that the mean is 72 and the standard deviation is 9, we need to calculate within how many standard deviations 45 and 99 fall:
- Distance of 45 from the mean = (45 - 72) / 9 = -3
- Distance of 99 from the mean = (99 - 72) / 9 = 3
Since 68% of the data falls within one standard deviation of the mean, we can expect that approximately 68% of the students scored between 45 and 99.
Therefore, if the distribution is assumed to be normal, approximately 68% of the students scored between 45 and 99, based on the Empirical Rule.
I hope this clarifies your confusion! If you have any more questions, feel free to ask.