For a particular sample of 53 scores on a psychology exam, the following results were obtained.
First quartile = 44 Third quartile = 68 Standard deviation = 8 Range = 55
Mean = 54 Median = 54 Mode = 71 Midrange = 64
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 30 and 78?
V. Assume that the distribution is normal. Based on the Empirical Rule, how many students scored between 22 and 86?
Please show all of your work.
I. 71. The mode, by definition is the answer that appears most frequently.
II. The midrange is 64, which is by definition the arithmetic mean of the largest and the smallest values in a sample or other group. The range is 55. Let H = highest score, and L = lowest score. Then,
(H+L)/2 = 64
H-L = 55
Use algebra to solve for H and L
III. See part II
IV
In general terms, the chebyshev’s theorem states that at least (1- 1/k2) of the elements of any distribution lie
Within k standard deviations of the mean (where k = a number greater than 1). 30 is 24 less than the mean of 54; 78 is 24 more than 54; 24 is 3 times the standard deviation of 8; So k = 3, so at least 1 - 1/9, or at least 8/9 of the total 53 scores lie between 30 and 78.
V. 22 is 4 standard deviations below the mean of 54, and 86 is 4 standard deviations above 54; According to the empirical rule, 68% of the values lie within 1 standard deviation of the mean, 95% of the values lie within 2 standard deviations of the mean, and 99.73% lie within 3 standard deviations of the mean. So even more than 99.73% would lie within 4 standard deviations of the mean; this number has to be a whole number, so the entire set of 53 scores lies within this range.
I. To determine the score that was earned by more students than any other score, we need to identify the mode of the dataset. The mode refers to the value that appears most frequently. In this case, the mode is given as 71. Therefore, the score earned by more students than any other score is 71.
II. The highest score earned on the exam can be determined by finding the maximum value in the dataset. The range of the dataset is given as 55, which represents the difference between the highest and lowest scores. Since the range is 55 and the lowest score is not provided, we can obtain the highest score by adding the range to the lowest score. Thus, the highest score is obtained by adding 55 to the lowest score.
III. The lowest score earned on the exam is not directly given in the information provided. However, we can calculate it by subtracting the range from the highest score. Using the formula from the previous step, the lowest score can be determined.
IV. According to Chebyshev's Theorem, we can determine the minimum proportion of scores that fall within a given number of standard deviations from the mean. The theorem states that at least (1 - (1 / k^2)) proportion of the data falls within k standard deviations of the mean. Here, k represents the number of standard deviations. In this case, the range is not used to calculate the required proportion. Therefore, only the mean and standard deviation are required to apply Chebyshev's Theorem.
To find the proportion of students scoring between 30 and 78, we need to calculate the number of standard deviations these values are from the mean. By subtracting the mean from both scores, you get the distance from the mean. Then divide each distance by the standard deviation to get the number of standard deviations.
V. Assuming a normal distribution, we can use the Empirical Rule to estimate the proportion of scores falling within certain ranges. 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.
To determine the number of students scoring between 22 and 86, we need to find the proportion of scores falling within that range using the formula:
Proportion = (Area under the curve between 22 and 86) = (Area to the right of 22) - (Area to the right of 86)
By converting the scores to z-scores (standardizing them in terms of standard deviations from the mean) and referring to a standard normal distribution table or using statistical software, we can find the areas under the curve and calculate the proportion.