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One​ year, the mean age of an inmate on death row was 38.4 years. A sociologist wondered whether the mean age of a​ death-row inmate has changed since then. She randomly selects 32 ​death-row inmates and finds that their mean age is 36.4​, with a standard deviation of 8.1. Construct a​ 95% confidence interval about the mean age. What does the interval​ imply?

Respuesta :

Answer:

95% Confidence interval:  (33.48,39.32)

Step-by-step explanation:

We are given the following in the question:

Population mean, μ =  38.4 years

Sample mean, [tex]\bar{x}[/tex] = 36.4 years

Sample size, n = 32

Alpha, α = 0.05

Sample standard deviation, s = 8.1 years

95% Confidence interval:  

[tex]\bar{x} \pm t_{critical}\displaystyle\frac{s}{\sqrt{n}}[/tex]  

Putting the values, we get,  

[tex]t_{critical}\text{ at degree of freedom 31 and}~\alpha_{0.05} = \pm 2.03[/tex]  

[tex]36.4 \pm 2.04(\frac{8.1}{\sqrt{32}} ) = 36.4 \pm 2.92 = (33.48,39.32)[/tex]  

Since the mean age age that is 38.4 years belongs to this 95% interval, there is enough evidence to say that the mean age  of an inmate on death row has not changed.