Assume that the article reported correct information. Complete the following sta
ID: 3175367 • Letter: A
Question
Assume that the article reported correct information. Complete the following statements about the distribution of systolic blood pressure measurements for women over seventy-five.
(a) According to Chebyshev's theorem, at least _________ of the measurements lie between 119.4 mmHg and 147.0.
(b) According to Chebyshev's theorem, at least 8/9 (about 89%) of the measurements lie between __________ mmHg and _________ mmHg. (Round your answer to 1 decimal place.)
(c) Suppose that the distribution is bell-shaped. According to the empirical rule, approximately 68% of the measurements lie between _________ mmHg and ________ mmHg.
(d) Suppose that the distribution is bell-shaped. According to the empirical rule, approximately __________ of the measurements lie between 119.4 mmHg and 147.0 .
133.2Explanation / Answer
Let X be the random variable that, systolic blood pressure measurement for women over seventy-five.
Given that, X has mean = 133.2 mmHg and standard deviation = 6.9 mmHg
Assume that the article reported correct information. Complete the following statements about the distribution of systolic blood pressure measurements for women over seventy-five.
a) Chebyshev's theorem states that
k = |(x - xbar) / s |
Let k>=1
Then the % of distirbution that lies within k SDs of the xbar is at least (1 - 1/k^2) *100.
We have given the interval is 119.4 mmHg to 147.0 mmHg.
Now first we have to find k then we can find %.
k for 119.4 and 147.0 :
k = |(119.4 - 133.2) / 6.9| = |-2| = 2
k = |(147.0 - 133.2) / 6.9 | = |2| = 2
SO here k=2 now we can use Chebyshev's theorem.
Percentage = (1 - 1/k^2)*100
= (1 - 1/2^2)*100
= 75%
According to Chebyshev's theorem, at least 75% of the measurements lie between 119.4 mmHg and 147.0.
b) Now in this part we have given percentage 8/9 (89%)
Percentage = 89% = 0.89
(1 - 1/k^2)*100 = 0.89
1 - 1/k^2 = 0.89/100
1 - 1/k^2 = 0.0089
1 - 0.0089 = 1/k^2
0.9911 = 1/k^2
k^2 = 1/0.9911 = 1.0
k = sqrt(1.01) = 1.0
lower limit = xbar - k*s = 133.2 - 1*6.9 = 126.3
upper limit = xbar + k*s = 133.2 + 1*6.9 = 140.1
According to Chebyshev's theorem, at least 8/9 (about 89%) of the measurements lie between 126.3 and 140.1.
c) Suppose that the distribution is bell-shaped.
Emperical rule states that 68% of the data lie within one standard deviation from the mean.
95% of the data lie within two standard deviation from the mean.
99.7% of the data lie within three standard deviation from the mean.
xbar - 1*s = 133.2 - 1*6.9 = 126.3
xbar + 1*s = 133.2 + 1*6.9 = 140.1
According to the empirical rule, approximately 68% of the measurements lie between 126.3 and 140.1 mmHg.
(d) Suppose that the distribution is bell-shaped.
We have to find k here.
k we can find by using same formula as in part a).
k = |(x-xbar) / s|
k for 119.4 and 147.0 are :
k = |(119.4 - 133.2)/6.9| = |-2| = 2
k = |(147.0 - 133.2) / 6.9| = |2| = 2
And by using emperical rule 95% of the data lie within two standard deviation from the mean.
According to the empirical rule, approximately 95% of the measurements lie between 119.4 mmHg and 147.0 .