Answer:
the 90% confidence interval is ( 48.684 , 51.316 )
Step-by-step explanation:
Given that :
the sample size = 36
Sample Mean = 50
standard deviation = 4.80
The objective is to calculate a 90% confidence interval.
At 90% confidence interval ;
the level of significance = 1 - 0.9 = 0.1
The critical value for [tex]z_{\alpha/2} = z_{0.1/2}[/tex]
[tex]= z_{0.05}[/tex] = 1.645
The standard error S.E = [tex]\dfrac{\sigma}{\sqrt{n}}[/tex]
=[tex]\dfrac{4.8}{\sqrt{36}}[/tex]
[tex]=\dfrac{4.8}{6}[/tex]
= 0.8
The Confidence interval level can be computed as:
[tex]\bar x \ \pm z \times \ \dfrac{ \sigma }{\sqrt{n}}[/tex]
For the lower limit :
[tex]\bar x \ - z \times \ \dfrac{ \sigma }{\sqrt{n}}[/tex]
[tex]=50 \ - 1.645 \times \ \dfrac{ 4.8 }{\sqrt{36}}[/tex]
[tex]=50 \ - 1.645 \times \ 0.8 }}[/tex]
=50 - 1.316
= 48.684
For the upper limit :
[tex]\bar x \ - z \times \ \dfrac{ \sigma }{\sqrt{n}}[/tex]
[tex]=50 \ + 1.645 \times \ \dfrac{ 4.8 }{\sqrt{36}}[/tex]
[tex]=50 \ + 1.645 \times \ 0.8 }}[/tex]
=50 + 1.316
= 51.316
Thus, the 90% confidence interval is ( 48.684 , 51.316 )
