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ID: 3325235 • Letter: 1
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 17 18 19 20 21 22 2324 251 Total tme remaining 128 Submit Test Question #16/ 25 An existing inventory for a test measuring self-esteem indicates that the scores have a standard deviation of 9. A psychologist gave the self-esteem test to a random sample of 80 individuals, and their mean score was 70, Construct a 95% confidence interval for the true mean of all test scores. Then complete the table below. Carry your intermediate computations to at least three decimal places. Round your answers to one decimal place (If necessary, consult a list of formulas.) what is the lower limit of the 95% confidence interval? what is the upper limit of the 95% confidence interval? Clear UndoExplanation / Answer
16
TRADITIONAL METHOD
given that,
standard deviation, =9
sample mean, x =70
population size (n)=80
I.
stanadard error = sd/ sqrt(n)
where,
sd = population standard deviation
n = population size
stanadard error = ( 9/ sqrt ( 80) )
= 1.006
II.
margin of error = Z a/2 * (stanadard error)
where,
Za/2 = Z-table value
level of significance, = 0.05
from standard normal table, two tailed z /2 =1.96
since our test is two-tailed
value of z table is 1.96
margin of error = 1.96 * 1.006
= 1.972
III.
CI = x ± margin of error
confidence interval = [ 70 ± 1.972 ]
= [ 68.028,71.972 ]
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DIRECT METHOD
given that,
standard deviation, =9
sample mean, x =70
population size (n)=80
level of significance, = 0.05
from standard normal table, two tailed z /2 =1.96
since our test is two-tailed
value of z table is 1.96
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 = [ 70 ± Z a/2 ( 9/ Sqrt ( 80) ) ]
= [ 70 - 1.96 * (1.006) , 70 + 1.96 * (1.006) ]
= [ 68.028,71.972 ]
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interpretations:
1. we are 95% sure that the interval [68.028 , 71.972 ] contains the true population mean
2. if a large number of samples are collected, and a confidence interval is created
for each sample, 95% of these intervals will contains the true population mean
[ANSWERS]
best point of estimate = mean = 70
standard error =1.006
z table value = 1.96
margin of error = 1.972
confidence interval = [ 68.028 , 71.972 ]
confidence interval =(68.0 , 71.9)
lower limit =68.028
upper limit = 71.972
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