Let [tex]\mu[/tex] be the population mean.
By analyzing the given information, we have the following hypothesis:-
[tex]H_0:\mu\geq81\\\\H_a:\mu<81[/tex]
Since the alternative hypotheses is left tailed so the test is a right-tailed test.
Also, the accuracy ratings of people in the general population are normally distributed.
Given : Sample size : n=51 , since n>30 so we use z-test.
Sample mean : [tex]\overline{x}=79[/tex]
Standard deviation : [tex]\sigma=\sqrt{20}\approx4.47[/tex]
Test statistic for population mean :-
[tex]z=\dfrac{\overline{x}-\mu}{\dfrac{\sigma}{\sqrt{n}}}[/tex]
[tex]\Rightarrow\ z=\dfrac{79-81}{\dfrac{4.47}{\sqrt{51}}}\\\\\Rightarrow\ z\approx-3.195[/tex]
Critical value (one-tailed) corresponds to the given significance level :-
[tex]z_{\alpha}=z_{0.05}=1.645[/tex]
Since the observed value of z (-3.195) is less than the critical value (1.645) , so we do not reject the null hypothesis.
Hence, we conclude that we do not have enough evidence to accept that people are less able to identify emotions correctly when they are extremely tired .
