Answer with explanation:
The 90% confidence interval (51%, 61%) for proportion means that the proportion of getting heads lie in it.
Given : Total number of times coin is tossed = 250
Number of times they got head =140
The probability of getting a head = 0.56
The confidence interval for proportion is given by :-
[tex]p\pm z_{\alpha/2}\sqrt{\dfrac{p(1-p)}{n}}[/tex]
Given significance level : [tex]\alpha=1-0.90=0.1[/tex]
Critical value : [tex]z_{\alpha/2}=z_{0.05}=\pm1.645[/tex]
Now, the 90​% confidence interval for proportion will be :-
[tex]0.56\pm (1.645)\sqrt{\dfrac{0.56(1-0.56)}{250}}\approx0.56\pm 0.0516\\\\=(0.56-0.0516,0.56+0.0516)=(0.5084,\ 0.6116)\approx(51\%,\ 61\%)[/tex]
Hence, the given confidence interval is correct.
