n a study designed to test the eflectiveness of magnets for treating back pain,

Question

n a study designed to test the eflectiveness of magnets for treating back pain, 40 patients were given a treatment with magnets and also a sham treatment without magnets Pain was measured uning a scale ton 0 (no pain) lo 100 (externe pan) Aner en the magr etteatteres, the 40 p tents had pain scores with 3mean et 110 and a stand d deuson of 2 SAher being g enthe sate int ds, he 40 paint, had p-con w m eur and a standand deviation of 2.1 Complete parts (a) through (c)belovw Click here to view atdstrbuion table Click here to ew pagn 2 of thestandard normal ditiution table .Construct the 90% cofdence itervaeslide otthe mean pain score tee paints gventhe magnet teatnert what is the coddence interval estimate ofthe popdation mean ? Round to orne decimal place as needed) b. Construct the 90% confidence interval estimabe of the mean pain soone for patients ghetsham beatment Round to one decimal place as needed)

Explanation / Answer

a.
TRADITIONAL METHOD
given that,
standard deviation, =2.5
sample mean, x =11
population size (n)=40
I.
stanadard error = sd/ sqrt(n)
where,
sd = population standard deviation
n = population size
stanadard error = ( 2.5/ sqrt ( 40) )
= 0.4
II.
margin of error = Z a/2 * (stanadard error)
where,
Za/2 = Z-table value
level of significance, = 0.1
from standard normal table, two tailed z /2 =1.645
since our test is two-tailed
value of z table is 1.645
margin of error = 1.645 * 0.4
= 0.7
III.
CI = x ± margin of error
confidence interval = [ 11 ± 0.7 ]
= [ 10.3,11.7 ]
-----------------------------------------------------------------------------------------------
DIRECT METHOD
given that,
standard deviation, =2.5
sample mean, x =11
population size (n)=40
level of significance, = 0.1
from standard normal table, two tailed z /2 =1.645
since our test is two-tailed
value of z table is 1.645
we use CI = x ± Z a/2 * (sd/ Sqrt(n))
where,
x = mean
sd = standard deviation
a = 1 - (confidence level/100)
Za/2 = Z-table value
CI = confidence interval
confidence interval = [ 11 ± Z a/2 ( 2.5/ Sqrt ( 40) ) ]
= [ 11 - 1.645 * (0.4) , 11 + 1.645 * (0.4) ]
= [ 10.3,11.7 ]
-----------------------------------------------------------------------------------------------
interpretations:
1. we are 90% sure that the interval [10.3 , 11.7 ] contains the true population mean
2. if a large number of samples are collected, and a confidence interval is created
for each sample, 90% of these intervals will contains the true population mean
[ANSWERS]
best point of estimate = mean = 11
standard error =0.4
z table value = 1.645
margin of error = 0.7
confidence interval = [ 10.3 , 11.7 ]

b.
TRADITIONAL METHOD
given that,
standard deviation, =2.1
sample mean, x =12.7
population size (n)=40
I.
stanadard error = sd/ sqrt(n)
where,
sd = population standard deviation
n = population size
stanadard error = ( 2.1/ sqrt ( 40) )
= 0.3
II.
margin of error = Z a/2 * (stanadard error)
where,
Za/2 = Z-table value
level of significance, = 0.1
from standard normal table, two tailed z /2 =1.645
since our test is two-tailed
value of z table is 1.645
margin of error = 1.645 * 0.3
= 0.5
III.
CI = x ± margin of error
confidence interval = [ 12.7 ± 0.5 ]
= [ 12.2,13.2 ]
-----------------------------------------------------------------------------------------------
DIRECT METHOD
given that,
standard deviation, =2.1
sample mean, x =12.7
population size (n)=40
level of significance, = 0.1
from standard normal table, two tailed z /2 =1.645
since our test is two-tailed
value of z table is 1.645
we use CI = x ± Z a/2 * (sd/ Sqrt(n))
where,
x = mean
sd = standard deviation
a = 1 - (confidence level/100)
Za/2 = Z-table value
CI = confidence interval
confidence interval = [ 12.7 ± Z a/2 ( 2.1/ Sqrt ( 40) ) ]
= [ 12.7 - 1.645 * (0.3) , 12.7 + 1.645 * (0.3) ]
= [ 12.2,13.2 ]
-----------------------------------------------------------------------------------------------
interpretations:
1. we are 90% sure that the interval [12.2 , 13.2 ] contains the true population mean
2. if a large number of samples are collected, and a confidence interval is created
for each sample, 90% of these intervals will contains the true population mean
[ANSWERS]
best point of estimate = mean = 12.7
standard error =0.3
z table value = 1.645
margin of error = 0.5
confidence interval = [ 12.2 , 13.2 ]

c.
do not overlap confidence intervals,it appears that the magnets treatments are less effective than sham treatments.

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