Question

# A random sample of 12 men aged 60 to 65 were selected and their blood pressures...

A random sample of 12 men aged 60 to 65 were selected and their blood pressures (in mmHg) were recorded.

127 135 118 164 143 130 125 153 120 173 140 180

a) Use your calculator to find the mean and standard deviation of their blood pressures.

b) Construct a 95% confidence interval for the standard deviation of the blood pressures of all men aged 60 to 65.

Solution :

a) We are given a data of sample size n = 12

127,135,118,164,143,130,125,153,120,173,140,180

Using this, first we find sample mean($url=https://private.codecogs.com/gif.latex?%5Cbar%20x&t=1596159762&ymreqid=69d777f6-ff7f-1f97-1cee-62000201a800&sig=7EoMV4JwgC.wHIj7JmbhdA--~D$) and sample standard deviation(s).

$url=https://private.codecogs.com/gif.latex?%5Cbar%20x&t=1596159762&ymreqid=69d777f6-ff7f-1f97-1cee-62000201a800&sig=7EoMV4JwgC.wHIj7JmbhdA--~D$ =   $url=https://private.codecogs.com/gif.latex?%5Cfrac%7Bsum%20of%20observations%7D%7Bn%7D&t=1596159762&ymreqid=69d777f6-ff7f-1f97-1cee-62000201a800&sig=Ub3HWh7C95NioCkAoxPPHg--~D$

= (127 + 135.......+ 180)/12

$url=https://private.codecogs.com/gif.latex?%5Cbar%20x&t=1596159762&ymreqid=69d777f6-ff7f-1f97-1cee-62000201a800&sig=7EoMV4JwgC.wHIj7JmbhdA--~D$ = 142.33

Now ,

s=   $url=https://private.codecogs.com/gif.latex?%5Csqrt%7B%5Cfrac%7B%5Csum_%7Bi%3D1%7D%5E%7Bn%7D%7B%28X_i%20-%20%5Coverline%7BX%7D%29%5E2%7D%20%7D%7Bn-1%7D%7D&t=1596159762&ymreqid=69d777f6-ff7f-1f97-1cee-62000201a800&sig=.kAbyGx8s3Cp.56ILuPWDQ--~D$

Using given data, find Xi - $url=https://private.codecogs.com/gif.latex?%5Cbar%20x&t=1596159762&ymreqid=69d777f6-ff7f-1f97-1cee-62000201a800&sig=7EoMV4JwgC.wHIj7JmbhdA--~D$ for each term.Take square for each.Then we can easily find s.

s = 20.847

b) Solution :

Given that,

c = 95% = 0.95

s = 20.847

n = 12

d.f. = n - 1 = 12 - 1 = 11

$url=http://latex.codecogs.com/gif.latex?%5Calpha&t=1596159732&ymreqid=69d777f6-ff7f-1f97-1cee-62000101a800&sig=OdfoXlY58K0QEpR4igsgKw--~D$ = 1 - 0.95 = 0.05

$url=http://latex.codecogs.com/gif.latex?%5Calpha&t=1596159732&ymreqid=69d777f6-ff7f-1f97-1cee-62000101a800&sig=OdfoXlY58K0QEpR4igsgKw--~D$ / 2 = 0.025

1 - ($url=http://latex.codecogs.com/gif.latex?%5Calpha&t=1596159732&ymreqid=69d777f6-ff7f-1f97-1cee-62000101a800&sig=OdfoXlY58K0QEpR4igsgKw--~D$ / 2) = 0.975

Now , using chi square table ,

$url=https://private.codecogs.com/gif.latex?%5Cchi_%7B%5Calpha/2%2Cdf%7D%5E%7B2%7D&t=1596159732&ymreqid=69d777f6-ff7f-1f97-1cee-62000101a800&sig=YaVdt_KSx5SJfCUzLwi3Qg--~D$ = $url=https://private.codecogs.com/gif.latex?%5Cchi%5E%7B2%7D&t=1596159732&ymreqid=69d777f6-ff7f-1f97-1cee-62000101a800&sig=_HMPuWUJ3_T48oniyzOyyA--~D$ 0.025,11 = 21.92

$url=https://private.codecogs.com/gif.latex?%5Cchi_%7B1%20-%20%28%5Calpha/2%29%2Cdf%7D%5E%7B2%7D&t=1596159732&ymreqid=69d777f6-ff7f-1f97-1cee-62000101a800&sig=HnBYQ6c5sT0viECzmXFq5A--~D$ =  $url=https://private.codecogs.com/gif.latex?%5Cchi%5E%7B2%7D&t=1596159732&ymreqid=69d777f6-ff7f-1f97-1cee-62000101a800&sig=_HMPuWUJ3_T48oniyzOyyA--~D$0.975,11 = 3.816

The 95% confidence interval for $url=https://private.codecogs.com/gif.latex?%5Csigma&t=1596159732&ymreqid=69d777f6-ff7f-1f97-1cee-62000101a800&sig=xs5J1EHF6P.aI4lZ.y9G5w--~D$ is,

$url=https://private.codecogs.com/gif.latex?s%5Csqrt%7B%5Cfrac%7Bn-1%7D%7B%5Cchi_%7B%5Calpha/2%2Cdf%7D%5E%7B2%7D%7D%7D%20%3C%5Csigma%3C%20s%5Csqrt%7B%5Cfrac%7Bn-1%7D%7B%5Cchi_%7B1%20-%20%5Calpha/2%2Cdf%7D%5E%7B2%7D%7D%7D&t=1596159732&ymreqid=69d777f6-ff7f-1f97-1cee-62000101a800&sig=4pSoY54xnb38kk1GScL7OQ--~D$

20.847  $url=https://private.codecogs.com/gif.latex?%5Csqrt%7B%7D&t=1596159732&ymreqid=69d777f6-ff7f-1f97-1cee-62000101a800&sig=EgIRy2QLsOP6ll9PvHrLng--~D$[(12 - 1) / 21.92]  < $url=https://private.codecogs.com/gif.latex?%5Csigma&t=1596159732&ymreqid=69d777f6-ff7f-1f97-1cee-62000101a800&sig=xs5J1EHF6P.aI4lZ.y9G5w--~D$ <  20.847$url=https://private.codecogs.com/gif.latex?%5Csqrt%7B%7D&t=1596159732&ymreqid=69d777f6-ff7f-1f97-1cee-62000101a800&sig=EgIRy2QLsOP6ll9PvHrLng--~D$[(12 - 1) / 3.816]

14.768 < $url=https://private.codecogs.com/gif.latex?%5Csigma&t=1596159732&ymreqid=69d777f6-ff7f-1f97-1cee-62000101a800&sig=xs5J1EHF6P.aI4lZ.y9G5w--~D$ < 35.396

The 95% confidence interval for $url=https://private.codecogs.com/gif.latex?%5Csigma&t=1596159732&ymreqid=69d777f6-ff7f-1f97-1cee-62000101a800&sig=xs5J1EHF6P.aI4lZ.y9G5w--~D$ is (14.768 , 35.396 )

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