I have read all my material regarding how to try and figure out how to use Chebyshev Theorem. I even email my teacher and to no avail, I need your help in how to work this problem out step by step.
A nationwide test taken by high school sophomores and juniors has three sections, each scored on a scale of 20 to 80 . In a recent year, the national mean score for the writing section was 48.0 , with a standard deviation of 10.0 . Based on this information, complete the following statements about the distribution of the scores on the writing section for the recent year.
A. According to Chebyshe's Theorem, at least __? (56% or 75% or 84%, or 89%) of the scores lie between 28.0 and 68.0.
b. According to Chevyshev's theorem, at least 36% of the scores lie betwen ___ and ___. (Round your answer to 1 deciaml place.)
C. Suppose that the distribution is bell-shaped. According to empiriacal rule, approximately 99.7% of the scores lie between __ and __.
D. Suppose that the distrubion is bell shaped. According to the empiriacal rule approximately __(68% or 75%or 95% or 99.7%) of the scores lie between 28.0 and 68.0%
To use Chebyshev's Theorem to estimate the proportion of scores lying within a certain range, you need to know the mean and standard deviation of the distribution.
In this case, the mean score on the writing section is given as 48.0, and the standard deviation is given as 10.0.
Let's go through each part of the problem step by step.
A. According to Chebyshev's Theorem, at least __? (56% or 75% or 84%, or 89%) of the scores lie between 28.0 and 68.0.
To find the proportion of scores that lie between two values, you need to calculate the range within a certain number of standard deviations from the mean. Chebyshev's Theorem states that for any distribution, at least (1 - (1/k^2)) * 100% of the data lie within k standard deviations of the mean, where k is any positive number greater than 1.
In this case, we want to find the range within 1 standard deviation from the mean, so k = 1. Using the formula, the proportion of scores within 1 standard deviation from the mean is at least (1 - (1/1^2)) * 100% = 0%. Thus, none of the scores lie between 28.0 and 68.0 according to Chebyshev's Theorem.
Therefore, the answer is 0%.
B. According to Chebyshev's Theorem, at least 36% of the scores lie between ___ and ___ (Round your answer to 1 decimal place.)
To find the range within which a certain proportion of scores lies, you need to solve for k in the Chebyshev's Theorem formula: (1 - (1/k^2)) * 100% = 36%.
Solving this equation gives us k^2 = 1 / (1 - 36%/100%) = 1 / 0.64 = 1.5625.
Taking the square root of both sides, we find k = √1.5625 ≈ 1.25.
To determine the range, we multiply the standard deviation (10.0) by the value of k:
Lower bound = mean - (k * standard deviation) = 48.0 - (1.25 * 10.0) = 48.0 - 12.5 = 35.5.
Upper bound = mean + (k * standard deviation) = 48.0 + (1.25 * 10.0) = 48.0 + 12.5 = 60.5.
Therefore, at least 36% of the scores lie between 35.5 and 60.5.
C. Suppose that the distribution is bell-shaped. According to the empirical rule, approximately 99.7% of the scores lie between __ and __.
The empirical rule applies to bell-shaped or approximately normal distributions. According to the empirical rule, approximately 99.7% of the scores lie within 3 standard deviations of the mean.
To determine the range, we multiply the standard deviation (10.0) by 3:
Lower bound = mean - (3 * standard deviation) = 48.0 - (3 * 10.0) = 48.0 - 30.0 = 18.0.
Upper bound = mean + (3 * standard deviation) = 48.0 + (3 * 10.0) = 48.0 + 30.0 = 78.0.
Therefore, approximately 99.7% of the scores lie between 18.0 and 78.0.
D. Suppose that the distribution is bell-shaped. According to the empirical rule, approximately __(68%, 75%, 95%, or 99.7%) of the scores lie between 28.0 and 68.0.
According to the empirical rule, approximately 68% of the scores lie within 1 standard deviation, approximately 95% of the scores lie within 2 standard deviations, and approximately 99.7% of the scores lie within 3 standard deviations.
Since the range between 28.0 and 68.0 is within 1 standard deviation of the mean, approximately 68% of the scores lie between these values.
Therefore, approximately 68% of the scores lie between 28.0 and 68.0.
To solve this problem step by step, we will use Chebyshev's Theorem and the empirical rule:
A. According to Chebyshev's Theorem, at least 75% of the scores lie between 28.0 and 68.0.
To find this answer:
1. Calculate the range of values within two standard deviations from the mean: (Mean - 2 * Standard Deviation) and (Mean + 2 * Standard Deviation).
(48.0 - 2 * 10.0) = 28.0 and (48.0 + 2 * 10.0) = 68.0
2. Use Chebyshev's Theorem, which states that at least (1 - 1/k^2) * 100% of the data lies within k standard deviations from the mean.
In this case, k = 2, so at least (1 - 1/2^2) * 100% = 75% of the scores lie between 28.0 and 68.0.
Therefore, the answer is 75%.
B. According to Chebyshev's theorem, at least 36% of the scores lie between (48 - k * 10) and (48 + k * 10). Round your answer to 1 decimal place.
To find this answer:
1. Set up the inequality: 36 <= (1 - 1/k^2) * 100%.
(1 - 1/k^2) >= 36/100
1 - 36/100 >= 1/k^2
64/100 <= 1/k^2
2. Solve for k: k >= sqrt(100/64) = sqrt(25/16) = 5/4 = 1.25
3. Substitute k into the equation from step 1: (48 - 1.25 * 10) and (48 + 1.25 * 10).
(48 - 1.25 * 10) = 35
(48 + 1.25 * 10) = 61.25 (rounded to 1 decimal place)
Therefore, at least 36% of the scores lie between 35 and 61.2.
C. Suppose that the distribution is bell-shaped. According to the empirical rule, approximately 99.7% of the scores lie between (Mean - 3 * Standard Deviation) and (Mean + 3 * Standard Deviation).
To find this answer:
1. Calculate the range of values within three standard deviations from the mean: (Mean - 3 * Standard Deviation) and (Mean + 3 * Standard Deviation).
(48.0 - 3 * 10.0) = 18.0 and (48.0 + 3 * 10.0) = 78.0
Therefore, approximately 99.7% of the scores lie between 18.0 and 78.0.
D. Suppose that the distribution is bell-shaped. According to the empirical rule, approximately 68% of the scores lie between 28.0 and 68.0.
Therefore, approximately 68% of the scores lie between 28.0 and 68.0.