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95% confident that the true population mean μ lies between 83.04 and 86.96
Confidence Interval
A confidence interval in statistics refers to the probability that a population parameter will fall between a set of values for a certain proportion of times. Analysts often use confidence intervals than contain either 95% or 99% of expected observations.
First need to determine whether dealing with means or proportions in this problem. The sample and population mean, that are dealing with means.
One sample mean, this mean creating a confidence interval for one sample (1 Sample t Interval)
Normally we would check for conditions, but since this is not formulated as a real-world scenario type problem, it is hard to check for randomness and independence. Therefore excluding conditions from this answer
Confidence Interval Formula
The formula for constructing a confidence interval for means is as follows:
x ± t ( σ / [tex]\sqrt{n}[/tex] )
The variables are:
x = 85
σ = 8
n = 64
Plug these values into the formula for the confidence interval
85 ± t ( 8 / [tex]\sqrt{64}[/tex])
Finding the Critical Value (t)
In order to find t, use this formula:
[tex]\frac{1 - C}{2} = A[/tex]
The z-score associated with A will give us t
So plug in the confidence interval 95% (.95) into the formula
[tex]\frac{1 - 0.95}{2} = 0.25[/tex]
Use the calculator or a t-table to find the z-score associated with this area under the curve
t = 1.96
Constructing Confidence Interval
Finish the confidence interval created
= 85 ± 1.96 ( 8 / [tex]\sqrt{64}[/tex] )
The confidence interval using this formula, to be
(83.04, 86.96)
Interpreting the Confidence Interval
95% confident that the true population mean μ lies between 83.04 and 86.96
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