For a particular sample of 63 scores on a psychology exam, the following results were obtained.
First quartile = 47 Third quartile = 81 Standard deviation = 11 Range = 68
Mean = 70 Median = 71 Mode = 77 Midrange = 55
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 48 and 92?
V. Assume that the distribution is normal. Based on the Empirical Rule, how many students scored between 59 and 81?
Please show all of your work.
I. To determine the score that was earned by more students than any other score, we need to find the mode of the sample. The mode represents the score that occurs with the highest frequency.
To find the mode, we look for the score that appears most frequently in the sample. Based on the information provided, the mode is 77. This means that more students earned a score of 77 than any other score.
II. The highest score earned on the exam can be determined by finding the range of the sample. The range represents the difference between the highest and lowest scores.
Given that the range is 68, we can calculate the highest score by adding the range to the lowest score. To find the lowest score, we can subtract the range from the highest score.
Let's denote the highest score as H and the lowest score as L. Therefore, H = L + Range.
From the information provided, the range is 68. Substituting the known values, we have H = L + 68. To solve for H, we need to find L. The value of L is not explicitly given in the information, but we can use the first quartile (Q1) to find it.
The first quartile (Q1) represents the score below which 25% of the data falls. In this case, Q1 = 47. Since the lowest score (L) falls below Q1, we can assume that the lowest score is smaller or equal to Q1.
Hence, the highest score (H) is Q1 + Range = 47 + 68 = 115.
Therefore, the highest score earned on the exam is 115.
III. The lowest score earned on the exam can be found by subtracting the range from the highest score (H) as mentioned in the previous explanation.
Hence, the lowest score (L) is H - Range = 115 - 68 = 47.
Therefore, the lowest score earned on the exam is 47.
IV. According to Chebyshev's Theorem, we can determine an approximate proportion of the scores that fall within a specific number of standard deviations from the mean.
Chebyshev's Theorem states that for any number k greater than 1, at least (1 - 1/k^2) of the data will fall within k standard deviations of the mean.
In this case, we are interested in knowing how many students scored between 48 and 92. Let's denote this range as R.
The mean of the sample is given as 70, and the standard deviation is given as 11. To calculate R, we need to determine the number of standard deviations away from the mean the scores of 48 and 92 are.
The formula to calculate the number of standard deviations away from the mean is (score - mean) / standard deviation.
For the lower bound of the range (48), the calculation is (48 - 70) / 11 = -22 / 11 = -2.
For the upper bound of the range (92), the calculation is (92 - 70) / 11 = 22 / 11 = 2.
Considering that Chebyshev's Theorem applies to any number k greater than 1, we can choose k = 2 to approximate the proportion of scores that fall between -2 and 2 standard deviations from the mean.
According to Chebyshev's Theorem, at least (1 - 1/k^2) of the data will fall within this range.
Using k = 2, we can calculate the approximate proportion of data within this range as 1 - 1/2^2 = 1 - 1/4 = 3/4 = 0.75.
Hence, according to Chebyshev's Theorem, at least 0.75 or 75% of the students scored between 48 and 92.
V. Assuming the distribution is normal, we can use the Empirical Rule (also known as the 68-95-99.7 rule) to estimate the proportion of data that falls within specific ranges based on the standard deviations from the mean.
The Empirical Rule states that for 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.
In this case, we are interested in knowing how many students scored between 59 and 81. Let's denote this range as R.
The mean of the sample is given as 70, and the standard deviation is given as 11.
To determine the proportion of students that fall within this range, we need to calculate the number of standard deviations away from the mean these scores are.
For the lower bound of the range (59), the calculation is (59 - 70) / 11 = -11 / 11 = -1.
For the upper bound of the range (81), the calculation is (81 - 70) / 11 = 11 / 11 = 1.
Since the range of -1 to 1 is within one standard deviation from the mean, we can apply the Empirical Rule and estimate that approximately 68% of the students scored between 59 and 81.
Therefore, based on the Empirical Rule, approximately 68% of the students scored between 59 and 81.