Answer:
The 95% confidence interval for the mean score is:
[tex]\mu[/tex] ∈ [tex](67.73,\ \ 84.67)[/tex]
Step-by-step explanation:
We want to estimate the population mean, that is, the average score of all the subjects.
An estimator for the population mean μ is [tex]{\displaystyle {\overline {x}}}[/tex]
Since the sample size is [tex]27 <30[/tex] then we use the statistic [tex]t_{\frac{\alpha}{2}}[/tex] with [tex]27-1 = 26[/tex] degrees of freedom
The confidence interval for the mean is:
[tex]({\displaystyle {\overline {x}}} + t_{\frac{\alpha}{2}}\frac{S}{\sqrt{n}},\ \ {\displaystyle {\overline {x}}} - t_{\frac{\alpha}{2}}\frac{S}{\sqrt{n}})[/tex]
Where
[tex]{\displaystyle {\overline {x}}}=76.2[/tex] is the average score of the sample
[tex]S=21.4[/tex] is the standard deviation of the sample
[tex]n= 27[/tex] is the sample size
[tex]t_{\frac{\alpha}{2}} = 2.056[/tex] for a confidence level of 95% and 26 degrees of freedom
Then confidence interval is
[tex](76.2 + 2.056\frac{21.4}{\sqrt{27}},\ \ {76.2 - 2.056\frac{21.4}{\sqrt{27}})[/tex]
[tex](76.2 + 8.467,\ \ 76.2 - 8.467)[/tex]
[tex]\mu[/tex] ∈ [tex](67.73,\ \ 84.67)[/tex]
