Question 1153956

 
a. What is the {{{95}}}% confidence interval estimate for {{{mu}}}?

For {{{95}}}% the {{{Z}}} value is {{{1.96}}}

use that {{{Z}}} in this formula for the Confidence Interval

{{{X}}}-  ±  {{{Z(delta/sqrt(n))}}}

Where:

    {{{X}}}-={{{36.5 }}} is the mean
    {{{Z=1.96}}}  is the chosen {{{Z}}}-value from the table  
    {{{delta=3}}} is the standard deviation
    {{{n=20}}}  is the number of observations

{{{36.5}}}  ±  {{{1.96(3/sqrt(20))}}}

{{{36.5 }}} ±  {{{1.96(0.67082)}}}
{{{36.5}}}  ±  {{{1.31}}}

solutions:

{{{36.5  +  1.31=37.8}}}
{{{36.5  - 1.31=35.2}}}

In other words: from {{{35.2}}} to {{{37.8}}}


b. Are the assumptions satisfied? Explain why.

A common assumption across all inferential tests is that the observations in your sample are independent from each other, meaning that the measurements for each sample subject are in no way influenced by or related to the measurements of other subjects.

Typical assumptions are:

    Normality: Data have a normal distribution (or at least is symmetric)
    Homogeneity of variances: Data from multiple groups have the same variance
    Linearity: Data have a linear relationship
    Independence: Data are independent

Therefore, your confidence interval applies to the sample mean, not the population mean. Ideally your data should be drawn from a normally distributed population. However, sample means of large numbers of observations tend to be distributed normally, whatever the underlying distribution.