When ? is unknown and the sample is of size n?30, there are two methods for computing confidence intervals for ?.
Method 1: Use the Student's t distribution with
d.f.= n?1.
This is the method used in the text. It is widely employed in
statistical studies. Also, most statistical software packages use
this method.
Method 2: When n ? 30, use the sample standard
deviation sas an estimate for ?, and then use the
standard normal distribution.
This method is based on the fact that for large samples,
sis a fairly good approximation for ?. Also, for
large n, the critical values for the Student's
tdistribution approach those of the standard normal
distribution.
(a) Now consider a sample size of 71, with sample mean x= 44.2, and sample standard deviation s = 6.2. Compute 90%, 95%, and 99% confidence intervals for ? using Method 1 with a Student's tdistribution. Round endpoints to two digits after the decimal.
90% | 95% | 99% | |
lower limit | |||
upper limit |
(b) Compute 90%, 95%, and 99% confidence intervals for
?using Method 2 with the standard normal distribution. Use
sas an estimate for ?. Round endpoints to two
digits after the decimal.
90% | 95% | 99% | |
lower limit | |||
upper limit |
a)here for (n-1=70) degree of freedom and 90,95 and 99% confidence inteval ; critical t are 1.667 ; 1.994 and 2.648
as confidence interval =sample mean -/+ t*std deviation/sqrt(n)
hence
90% | 95% | 99% | |
lower limit | 42.97 | 42.73 | 42.25 |
upper limit | 45.43 | 45.67 | 46.15 |
b)
for 90,95 and 99% confidence inteval ; critical z are 1.645 ; 1.96 and 2.58
confidence limit =sample mean -/+ z*std deviation/sqrt(n)
90% | 95% | 99% | |
lower limit | 42.99 | 42.76 | 42.30 |
upper limit | 45.41 | 45.64 | 46.10 |
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