We have to find the margin of error at a 90% confidence level for estimating the population mean.
The sample size is n = 22, the sample mean is M = 35 and the sample standard deviation is s = 3.
In this case, we don't know the population standard deviation (σ). We have to estimate it from the sample standard deviation.
To do that, we use the Student's t distribution instead of the normal distribution.
When σ is not known, s divided by the square root of N is used as an estimate of σ:
[tex]s_M=\dfrac{s}{\sqrt{N}}=\dfrac{3}{\sqrt{22}}=\dfrac{3}{4.69}=0.64[/tex]
The degrees of freedom for this sample size are:
[tex]df=n-1=22-1=21[/tex]
The t-value for a 90% confidence interval and 21 degrees of freedom is t = 1.721.
The margin of error (MOE) can be calculated as:
[tex]MOE=t\cdot s_M=1.721\cdot0.64\approx1.10[/tex]
Answer: the margin of error is 1.10.
