For a particular sample of 63 scores on a psychology exam, the following results were obtained.
First quartile = 57 Third quartile = 72 Standard deviation = 9 Range = 52
Mean = 72 Median = 68 Mode = 70 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 54 and 90?
V. Assume that the distribution is normal. Based on the Empirical Rule, how many students scored between 54 and 90?
Please show all work!
I'll give you a few hints.
For the first three questions, you should be able to look at the data given and determine the values. (For example, remember that the mode is the most frequently occurring score in a distribution.)
For the last two questions, see below.
Chebyshev's Theorem says:
1. Within two standard deviations of the mean, you will find at least 75% of the data.
2. Within three standard deviations of the mean, you will find at least 89% of the data.
Empirical Rule says:
1. Within two standard deviations of the mean, you will find about 95% of the data.
2. Within three standard deviations of the mean, you will find about 99.7% of the data.
From the data given:
Mean = 72, Standard deviation = 9
Therefore, between 54 and 90 fall within 2 standard deviations about the mean. (72 - 18 = 54; 72 + 18 = 90)
To answer your last two questions, take the appropriate percentages and multiply by the sample size of 63 to get the number of students between 54 and 90.
I hope this will help.
I. To determine the score earned by more students than any other score (mode), you need to look at the given information. The mode is the score that appears most frequently in the data. In this case, the mode is 70, which means that more students earned a score of 70 than any other score.
II. The highest score earned on the exam can be determined by finding the maximum value in the data set or using the range. In this case, the range is given as 52, so you can add this value to the lowest score to find the highest score. The lowest score is not explicitly given in the information provided, so you'll need to find it using other available data.
III. The lowest score earned on the exam is not explicitly given in the information provided. To find the lowest score, you can use the range. The range is the difference between the highest and lowest values in the data set. In this case, the range is given as 52, so you can subtract this value from the highest score to find the lowest score.
IV. According to Chebyshev's Theorem, you can determine the proportion of scores that fall within a certain number of standard deviations from the mean. The theorem states that at least (1 - 1/k^2) of the data will fall within k standard deviations of the mean, where k is any positive integer greater than 1. In this case, the standard deviation is given as 9, and you want to find the number of students who scored between 54 and 90. To apply Chebyshev's Theorem, you need to determine the number of standard deviations between the mean and the lower and upper bounds. First, calculate the difference between the mean and the lower bound (54 - 72 = -18), and then divide this value by the standard deviation to find the number of standard deviations. Similarly, calculate the difference between the mean and the upper bound (90 - 72 = 18) and divide by the standard deviation. Keep in mind that this theorem provides a lower bound, so there could be more students within the given range. So, you can use k = 2, which means that at least (1 - 1/2^2) = 75% of the data will fall within 2 standard deviations of the mean. Multiply this proportion by the total number of students (63) to get the approximate number of students between 54 and 90.
V. Assuming the distribution is normal, you can use the Empirical Rule (also known as the 68-95-99.7 rule) to estimate the number of students who scored between 54 and 90. The Empirical Rule states that 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. In this case, the standard deviation is given as 9. By considering the mean and the number of standard deviations from the mean, you can estimate the proportion of students between 54 and 90. Then, multiply this proportion by the total number of students (63) to get the approximate number of students within the given range.
It's important to note that the actual distribution may not be perfectly normal, as these calculations assume. However, these rules provide rough estimates based on the given information.