for a particular sample of 77 scores on a psychology exam, the following results were obtained:
first quartile=44, third quartile= 71, standard deviation=6, range=45, mean=64,median=57, mode=32, midrange=60
a)according to chevyshev's theorem how many students scored between 48 and 88?
b) assume that the distribution is normal. based on the empirical rule, how many students scored between 46 and 82?
* i came up with 12 for a and 24 for b, is that correct?
please help and show where i went wrong. thank you
From the data, mode = 32, median = 57 and mean 64, the distribution is definitely positively skewed (to the right).
Unfortunately, I don't know Chevyshev's theorem, and I would not assume the distribution to be normal.
However, for a normal distribution, Z = (score-mean)/SD
Find the Z scores for 46 and 82, then Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions related to those Z scores and multiply those proportions by 77.
To answer these questions, we'll use Chebyshev's theorem and the empirical rule. Let's go through each question step by step:
a) According to Chebyshev's theorem, we can determine the proportion of data within a certain number of standard deviations from the mean for any distribution, regardless of shape. Chebyshev's theorem states that at least (1 - 1/k^2) of the data falls within k standard deviations of the mean, where k is any positive constant greater than 1.
In this case, we want to find the proportion of students who scored between 48 and 88. To do this, we need to calculate the number of standard deviations away from the mean these scores are.
The mean is given as 64, and the standard deviation is given as 6. For a score of 48:
Number of standard deviations away from the mean = (48 - mean) / standard deviation
Number of standard deviations away from the mean = (48 - 64) / 6 = -16 / 6 = -2.67
For a score of 88:
Number of standard deviations away from the mean = (88 - mean) / standard deviation
Number of standard deviations away from the mean = (88 - 64) / 6 = 24 / 6 = 4
Now, using Chebyshev's theorem, we calculate the proportion of data that falls within 2.67 and 4 standard deviations away from the mean:
Proportion of data for k = 2.67: 1 - 1/2.67^2 = 1 - 1/7.1289 ≈ 0.866
Proportion of data for k = 4: 1 - 1/4^2 = 1 - 1/16 = 15/16 ≈ 0.938
The proportion of students who scored between 48 and 88 is the difference between these two proportions:
Proportion of data between 48 and 88 = 0.938 - 0.866 = 0.072
To find the number of students, multiply this proportion by the total number of students (77):
Number of students = Proportion of data between 48 and 88 * Total number of students
Number of students = 0.072 * 77 = 5.544 ≈ 6 (rounded to the nearest whole number)
Therefore, according to Chebyshev's theorem, around 6 students scored between 48 and 88.
b) The empirical rule, also known as the 68-95-99.7 rule, applies specifically to normally distributed data. It 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, we want to find the proportion of students who scored between 46 and 82 based on the assumption that the distribution is normal. We can use the empirical rule to estimate this.
First, calculate the number of standard deviations away from the mean these scores are, using the mean of 64 and the standard deviation of 6:
For a score of 46:
Number of standard deviations away from the mean = (46 - mean) / standard deviation
Number of standard deviations away from the mean = (46 - 64) / 6 = -18 / 6 = -3
For a score of 82:
Number of standard deviations away from the mean = (82 - mean) / standard deviation
Number of standard deviations away from the mean = (82 - 64) / 6 = 18 / 6 = 3
Since the empirical rule tells us that approximately 95% of the data falls within two standard deviations of the mean, we can use this information to estimate the proportion of students who scored between 46 and 82.
Proportion of data = 0.95
To find the number of students, multiply this proportion by the total number of students (77):
Number of students = Proportion of data between 46 and 82 * Total number of students
Number of students = 0.95 * 77 = 73.15 ≈ 73 (rounded to the nearest whole number)
Therefore, based on the assumption of a normal distribution using the empirical rule, approximately 73 students scored between 46 and 82.
Your answers are different from the correct ones. For part a), the correct answer is around 6 students, and for part b), the correct answer is around 73 students.